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# Chapter 8. Random Vectors and joint Distributions

## 8.1. Random Vectors and Joint Distributions^{*}

### Introduction

A single, real-valued random variable is a function
(mapping) from the basic space *Ω* to the real line. That is, to each possible outcome *ω*
of an experiment there corresponds a real value *t*=*X*(*ω*). The mapping induces a
probability mass distribution on the real line, which provides a means of making probability
calculations. The distribution is described by a distribution function *F _{X}*. In the
absolutely continuous case, with no point mass concentrations, the distribution may also be
described by a probability density function

*f*. The probability density is the linear density of the probability mass along the real line (i.e., mass per unit length). The density is thus the derivative of the distribution function. For a simple random variable, the probability distribution consists of a point mass

_{X}*p*at each possible value

_{i}*t*of the random variable. Various m-procedures and m-functions aid calculations for simple distributions. In the absolutely continuous case, a simple approximation may be set up, so that calculations for the random variable are approximated by calculations on this simple distribution.

_{i} Often we have more than one random variable. Each can be considered separately, but
usually they have some probabilistic ties which must be taken into account when they
are considered jointly. We treat the joint case by considering the individual random
variables as *coordinates of a random vector*. We extend the techniques for a single
random variable to the multidimensional case. To simplify exposition and to keep calculations
manageable, we consider a pair of random variables as coordinates of a two-dimensional
random vector. The concepts and results extend directly to any finite number of random
variables considered jointly.

### Random variables considered jointly; random vectors

As a starting point, consider a simple example in which the probabilistic interaction between two random quantities is evident.

Two campus jobs are open. Two juniors and three seniors apply. They seem equally
qualified, so it is decided to select them by chance. Each combination of two is
equally likely. Let *X* be the number of juniors selected (possible values 0, 1, 2) and
*Y* be the number of seniors selected (possible values 0, 1, 2). However there are only
three possible pairs of values for , or . Others
have zero probability, since they are impossible. Determine the probability for each of the
possible pairs.

**SOLUTION**

There are equally likely pairs. Only one pair can be both juniors. Six pairs can be one of each. There are ways to select pairs of seniors. Thus

These probabilities add to one, as they must, since this exhausts the mutually exclusive possibilities. The probability of any other combination must be zero. We also have the distributions for the random variables conisidered individually.

We thus have a *joint distribution* and two individual or *marginal distributions*.

We formalize as follows:

A pair of random variables considered jointly is treated as the pair
of coordinate functions for a two-dimensional random vector . To each *ω*∈*Ω*, *W* assigns the pair of real numbers , where
*X*(*ω*)=*t* and *Y*(*ω*)=*u*. If we represent the pair of values as the point
on the plane, then , so that

is a mapping from the basic space *Ω* to the plane *R ^{2}*. Since

*W*is a function, all mapping ideas extend. The inverse mapping

*W*

^{–1}plays a role analogous to that of the inverse mapping

*X*

^{–1}for a real random variable. A two-dimensional vector

*W*is a

*random vector*iff

*W*

^{–1}(

*Q*) is an event for each reasonable set (technically, each Borel set) on the plane.

A fundamental result from measure theory ensures

is a random vector iff each of the coordinate functions *X* and *Y* is a random variable.

In the selection example above, we model (the number of juniors selected) and
*Y* (the number of seniors selected) as random variables. Hence the vector-valued
function

### Induced distribution and the joint distribution function

In a manner parallel to that for the single-variable case, we obtain a mapping of
probability mass from the basic space to the plane. Since *W*^{–1}(*Q*) is an event
for each reasonable set *Q* on the plane, we may assign to *Q* the probability mass

Because of the preservation of set operations by inverse mappings as in the single-variable case,
the mass assignment determines
*P*_{XY} as a probability measure on the subsets of the plane *R ^{2}*. The argument
parallels that for the single-variable case. The result is the

*probability distribution induced by*. To determine the probability that the vector-valued function takes on a (vector) value in region

*Q*, we simply determine how much induced probability mass is in that region.

To determine , we determine the region for which the
first coordinate value (which we call *t*) is between one and three and the second
coordinate value (which we call *u*) is greater than zero. This corresponds to
the set *Q* of points on the plane with 1≤*t*≤3 and *u*>0. Gometrically,
this is the strip on the plane bounded by (but not including) the horizontal axis and by the
vertical lines *t*=1 and *t*=3 (included). The problem is to determine how much
probability mass lies in that strip. How this is acheived depends upon the nature of the
distribution and how it is described.

As in the single-variable case, we have a distribution function.

**Definition**

The *joint distribution function **F*_{XY} for is
given by

This means that is equal to the probability mass in the region *Q*_{tu}
on the plane such that the first coordinate is less than or equal to *t* and the second
coordinate is less than or equal to *u*. Formally, we may write

Now for a given point (*a*,*b*), the region *Q*_{ab} is the set of points (*t*,*u*) on
the plane which are on or to the left of the vertical line through (*t*,0)*and*
on or below the horizontal line through (0,*u*) (see Figure 1 for specific point *t*=*a*,*u*=*b*).
We refer to such regions as semiinfinite intervals on the plane.

The theoretical result quoted in the real variable case extends to ensure that a distribution on the
plane is determined uniquely by consistent assignments to the semiinfinite intervals
*Q*_{tu}. Thus, the induced distribution is determined completely by the joint
distribution function.

**Distribution function for a discrete random vector**

The induced distribution consists of point masses. At point in the range of there is probability mass . As in the general case, to determine we determine how much probability mass is in the region. In the discrete case (or in any case where there are point mass concentrations) one must be careful to note whether or not the boundaries are included in the region, should there be mass concentrations on the boundary.

The probability distribution is quite simple. Mass 3/10 at (0,2), 6/10 at (1,1), and
1/10 at (2,0). This distribution is plotted in Figure 8.2. To determine (and visualize)
the joint distribution function, think of moving the point on the plane. The
region *Q*_{tu} is a giant “sheet” with corner at . The value of
is the amount of probability covered by the sheet. This value is constant over any grid
cell, including the left-hand and lower boundariies, and is the value taken on at the lower
left-hand corner of the cell. Thus, if is in any of the
three squares on the lower left hand part of the diagram, no probability mass is covered
by the sheet with corner in the cell. If is on or in the square having probability
6/10 at the lower left-hand corner, then the sheet covers that probability, and the value of
. The situation in the other cells may be checked out by this procedure.

**Distribution function for a mixed distribution**

The pair produces a mixed distribution as follows (see Figure 8.3)

Point masses 1/10 at points (0,0), (1,0), (1,1), (0,1)

Mass 6/10 spread uniformly over the unit square with these vertices

The joint distribution function is zero in the second, third, and fourth quadrants.

If the point is in the square or on the left and lower boundaries, the sheet covers the point mass at (0,0) plus 0.6 times the area covered within the square. Thus in this region

(8.7)If the pont is above the square (including its upper boundary) but to the left of the line

*t*=1, the sheet covers two point masses plus the portion of the mass in the square to the left of the vertical line through . In this case(8.8)If the point is to the right of the square (including its boundary) with 0≤

*u*<1, the sheet covers two point masses and the portion of the mass in the square below the horizontal line through , to give(8.9)If is above and to the right of the square (i.e., both 1≤

*t*and 1≤*u*). then all probability mass is covered and in this region.

### Marginal distributions

If the joint distribution for a random vector is known, then the distribution for each of the
component random variables may be determined. These are known as *marginal distributions*. In general, the converse is not true. However, if the component random variables form an
independent pair, the treatment in that case shows that the marginals determine the joint
distribution.

To begin the investigation, note that

Thus

This may be interpreted with the aid of Figure 8.4. Consider the sheet for point .

If we push the point up vertically, the upper boundary of *Q*_{tu} is pushed up until eventually
all probability mass on or to the left of the vertical line through is included. This
is the total probability that *X*≤*t*. Now *F*_{X}(*t*) describes probability mass on the line.
The probability mass described by *F*_{X}(*t*) is the same as the total joint probability mass on
or to the left of the vertical line through . We may think of the mass in the half
plane being projected onto the horizontal line to give the *marginal* distribution for *X*.
A parallel argument holds for the marginal for *Y*.

This mass is projected onto the vertical axis to give the marginal distribution for *Y*.

**Marginals for a joint discrete distribution**

Consider a joint simple distribution.

Thus, all the probability mass on the vertical line through is projected onto
the point *t _{i}* on a horizontal line to give . Similarly, all the probability
mass on a horizontal line through is projected onto the point

*u*on a vertical line to give .

_{j}The pair produces a joint distribution that places mass 2/10 at each of the five points

(See Figure 8.5)

The marginal distribution for *X* has masses 2/10, 2/10, 4/10, 2/10 at points
*t*=0,1,2,3, respectively. Similarly, the marginal distribution for *Y*
has masses 4/10, 4/10, 2/10 at points *u*=0,1,2, respectively.

Consider again the joint distribution in Example 8.4. The pair produces a mixed distribution as follows:

Point masses 1/10 at points (0,0), (1,0), (1,1), (0,1)

Mass 6/10 spread uniformly over the unit square with these vertices

The construction in Figure 8.6 shows the graph of the marginal distribution function *F _{X}*. There is a jump in the amount of 0.2 at

*t*=0, corresponding to the two point masses on the vertical line. Then the mass increases linearly with

*t*, slope 0.6, until a final jump at

*t*=1 in the amount of 0.2 produced by the two point masses on the vertical line. At

*t*=1, the total mass is “covered” and

*F*

_{X}(

*t*) is constant at one for

*t*≥1. By symmetry, the marginal distribution for

*Y*is the same.

## 8.2. Random Vectors and MATLAB^{*}

### m-procedures for a pair of simple random variables

We examine, first, calculations on a pair of simple random variables *X*,*Y*, considered jointly.
These are, in effect, two components of a random vector , which maps from
the basic space *Ω* to the plane. The induced distribution is on the
-plane. Values on the horizontal axis (*t*-axis) correspond to values of
the first coordinate random variable *X* and values on the vertical axis (*u*-axis)
correspond to values of *Y*. We extend the computational strategy used for a single
random variable.

First, let us review the one-variable strategy. In this case, data consist of values *t _{i}*
and corresponding probabilities arranged in matrices

To perform calculations on *Z*=*g*(*X*), we we use array operations on *X* to form a matrix

which has in a position corresponding to
in matrix **PX**.

*Basic problem*. Determine *P*(*g*(*X*)∈*M*), where *M* is some prescribed set of
values.

Use relational operations to determine the

*positions*for which . These will be in a zero-one matrix*N*, with ones in the desired positions.Select the in the corresponding positions and sum. This is accomplished by one of the MATLAB operations to determine the inner product of

*N*and**PX**

We extend these techniques and strategies to a pair of simple random variables, considered jointly.

The data for a pair {

*X*,*Y*} of random variables are the values of*X*and*Y*, which we may put in row matrices(8.16)and the joint probabilities in a matrix

*P*. We usually represent the distribution graphically by putting probability mass at the point on the plane. This joint probability may is represented by the matrix*P*with elements arranged corresponding to the mass points on the plane. Thus(8.17)To perform calculations, we form computational matrices

*t*and*u*such that —*t*has element*t*at each position (i.e., at each point on the_{i}*i*th column from the left) —*u*has element*u*at each position (i.e., at each point on the_{j}*j*th row from the bottom) MATLAB array and logical operations on*t*,*u*,*P*perform the specified operations on*t*_{i},*u*_{j}, and at each position, in a manner analogous to the operations in the single-variable case.Formation of the

*t*and*u*matrices is achieved by a basic setup m-procedure called*jcalc*. The data for this procedure are in three matrices: is the set of values for random variable*X*is the set of values for random variable*Y*, and , where . We arrange the joint probabilities as on the plane, with*X*-values increasing to the right and*Y*-values increasing upward. This is different from the usual arrangement in a matrix, in which values of the second variable increase downward. The m-procedure takes care of this inversion. The m-procedure forms the matrices*t*and*u*, utilizing the MATLAB function*meshgrid*, and computes the marginal distributions for*X*and*Y*. In the following example, we display the various steps utilized in the setup procedure. Ordinarily, these intermediate steps would not be displayed.Example 8.7. Setup and basic calculations>> jdemo4 % Call for data in file jdemo4.m >> jcalc % Call for setup procedure Enter JOINT PROBABILITIES (as on the plane) P Enter row matrix of VALUES of X X Enter row matrix of VALUES of Y Y Use array operations on matrices X, Y, PX, PY, t, u, and P >> disp(P) % Optional call for display of P 0.0360 0.0198 0.0297 0.0209 0.0180 0.0372 0.0558 0.0837 0.0589 0.0744 0.0516 0.0774 0.1161 0.0817 0.1032 0.0264 0.0270 0.0405 0.0285 0.0132 >> PX % Optional call for display of PX PX = 0.1512 0.1800 0.2700 0.1900 0.2088 >> PY % Optional call for display of PY PY = 0.1356 0.4300 0.3100 0.1244 - - - - - - - - - - % Steps performed by jcalc >> PX = sum(P) % Calculation of PX as performed by jcalc PX = 0.1512 0.1800 0.2700 0.1900 0.2088 >> PY = fliplr(sum(P')) % Calculation of PY (note reversal) PY = 0.1356 0.4300 0.3100 0.1244 >> [t,u] = meshgrid(X,fliplr(Y)); % Formation of t, u matrices (note reversal) >> disp(t) % Display of calculating matrix t -3 0 1 3 5 % A row of X-values for each value of Y -3 0 1 3 5 -3 0 1 3 5 -3 0 1 3 5 >> disp(u) % Display of calculating matrix u 2 2 2 2 2 % A column of Y-values (increasing 1 1 1 1 1 % upward) for each value of X 0 0 0 0 0 -2 -2 -2 -2 -2

Suppose we wish to determine the probability . Using array operations on

*t*and*u*, we obtain the matrix .>> G = t.^2 - 3*u % Formation of G = [g(t_i,u_j)] matrix G = 3 -6 -5 3 19 6 -3 -2 6 22 9 0 1 9 25 15 6 7 15 31 >> M = G >= 1 % Positions where G >= 1 M = 1 0 0 1 1 1 0 0 1 1 1 0 1 1 1 1 1 1 1 1 >> pM = M.*P % Selection of probabilities pM = 0.0360 0 0 0.0209 0.0180 0.0372 0 0 0.0589 0.0744 0.0516 0 0.1161 0.0817 0.1032 0.0264 0.0270 0.0405 0.0285 0.0132 >> PM = total(pM) % Total of selected probabilities PM = 0.7336 % P(g(X,Y) >= 1)

In Example 3 from "Random Vectors and Joint Distributions" we note that the joint distribution function

*F*_{XY}is constant over any grid cell, including the left-hand and lower boundaries, at the value taken on at the lower left-hand corner of the cell. These lower left-hand corner values may be obtained systematically from the joint probability matrix*P*by a two step operation.Take cumulative sums upward of the columns of

*P*.Take cumulative sums of the rows of the resultant matrix.

This can be done with the MATLAB function cumsum, which takes column cumulative sums downward. By flipping the matrix and transposing, we can achieve the desired results.

Example 8.8. Calculation of*F*_{XY}values for Example 3 from "Random Vectors and Joint Distributions">> P = 0.1*[3 0 0; 0 6 0; 0 0 1]; >> FXY = flipud(cumsum(flipud(P))) % Cumulative column sums upward FXY = 0.3000 0.6000 0.1000 0 0.6000 0.1000 0 0 0.1000 >> FXY = cumsum(FXY')' % Cumulative row sums FXY = 0.3000 0.9000 1.0000 0 0.6000 0.7000 0 0 0.1000

Figure 8.7.Comparison with Example 3 from "Random Vectors and Joint Distributions" shows agreement with values obtained by hand.

The two step procedure has been incorprated into an m-procedure*jddbn*. As an example, return to the distribution in Example Example 8.7

Example 8.9. Joint distribution function for Example 8.7>> jddbn Enter joint probability matrix (as on the plane) P To view joint distribution function, call for FXY >> disp(FXY) 0.1512 0.3312 0.6012 0.7912 1.0000 0.1152 0.2754 0.5157 0.6848 0.8756 0.0780 0.1824 0.3390 0.4492 0.5656 0.0264 0.0534 0.0939 0.1224 0.1356

These values may be put on a grid, in the same manner as in Figure 2 for Example 3 in "Random Vectors and Joint Distributions".

As in the case of canonic for a single random variable, it is often useful to have a function version of the procedure jcalc to provide the freedom to name the outputs conveniently.

*function*`[x,y,t,u,px,py,p] = jcalcf(X,Y,P)`

The quantities*x*,*y*,*t*,*u*,*p**x*,*p**y*, and*p*may be given any desired names.

### Joint absolutely continuous random variables

In the single-variable case, the condition that there are no point mass concentrations on the line ensures the existence of a probability density function, useful in probability calculations. A similar situation exists for a joint distribution for two (or more) variables. For any joint mapping to the plane which assigns zero probability to each set with zero area (discrete points, line or curve segments, and countable unions of these) there is a density function.

**Definition**

If the joint probability distribution for the pair {*X*,*Y*} assigns
zero probability to every set of points with zero area, then there exists a *joint density
function* *f*_{XY} with the property

We have three properties analogous to those for the single-variable case:

At every continuity point for *f*_{XY}, the density is the second partial

Now

A similar expression holds for *F*_{Y}(*u*). Use of the fundamental theorem of calculus to
obtain the derivatives gives the result

*Marginal densities*. Thus, to obtain the marginal density for the first variable,
integrate out the second variable
in the joint density, and similarly for the marginal for the second variable.

Let . This region is the triangle
bounded by *u*=0, *u*=*t*, and *t*=1 (see Figure 8.8)

where *Q* is the common part of the
triangle with the strip between *t*=0.5 and *t*=0.75 and above the line *u*=0.5. This is the small triangle bounded by *u*=0.5, *u*=*t*, and *t*=0.75. Thus

The pair has joint density
on the region bounded by *t*=0, *t*=2, *u*=0, and (see Figure 8.9).
Determine the marginal density *f _{X}*.

SOLUTION

Examination of the figure shows that we have different limits for the integral with respect
to *u* for 0≤*t*≤1 and for 1<*t*≤2.

For 0≤

*t*≤1(8.26)For 1<

*t*≤2(8.27)

We may combine these into a single expression in a manner used extensively in subsequent
treatments. Suppose and . Then *I*_{M}(*t*)=1 for *t*∈*M*
(i.e., 0≤*t*≤1) and zero elsewhere. Likewise, *I*_{N}(*t*)=1 for *t*∈*N* and
zero elsewhere. We can, therefore express *f _{X}* by

### Discrete approximation in the continuous case

For a pair with joint density *f*_{XY}, we approximate the distribution
in a manner similar to that for a single random variable. We then utilize the techniques
developed for a pair of simple random variables. If we have *n* approximating values
*t _{i}* for

*X*and

*m*approximating values

*u*for

_{j}*Y*, we then have

*n*·

*m*pairs , corresponding to points on the plane. If we subdivide the horizontal axis for values of

*X*, with constant increments

**dx**, as in the single-variable case, and the vertical axis for values of

*Y*, with constant increments

**dy**, we have a grid structure consisting of rectangles of size

*d*

*x*·

*d*

*y*. We select

*t*and

_{i}*u*at the midpoint of its increment, so that the point is at the midpoint of the rectangle. If we let the approximating pair be , we assign

_{j}As in the one-variable case, if the increments are small enough,

The m-procedure *tuappr* calls for endpoints of intervals which include the
ranges of *X* and *Y* and for the numbers of subintervals on each. It then
prompts for an expression for , from which it determines
the joint probability distribution. It calculates the marginal approximate
distributions and sets up the calculating matrices *t* and *u* as does the
m-process jcalc for simple random variables. Calculations are then carried out
as for any joint simple pair.

Determine .

>> tuappr Enter matrix [a b] of X-range endpoints [0 1] Enter matrix [c d] of Y-range endpoints [0 1] Enter number of X approximation points 200 Enter number of Y approximation points 200 Enter expression for joint density 3*(u <= t.^2) Use array operations on X, Y, PX, PY, t, u, and P >> M = (t <= 0.8)&(u > 0.1); >> p = total(M.*P) % Evaluation of the integral with p = 0.3355 % Maple gives 0.3352455531

The discrete approximation may be used to obtain approximate plots of marginal distribution and density functions.

on the triangle bounded by *u*=0, *u*≤1+*t*, and
*u*≤1–*t*.

>> tuappr Enter matrix [a b] of X-range endpoints [-1 1] Enter matrix [c d] of Y-range endpoints [0 1] Enter number of X approximation points 400 Enter number of Y approximation points 200 Enter expression for joint density 3*u.*(u<=min(1+t,1-t)) Use array operations on X, Y, PX, PY, t, u, and P >> fx = PX/dx; % Density for X (see Figure 8.10) % Theoretical (3/2)(1 - |t|)^2 >> fy = PY/dy; % Density for Y >> FX = cumsum(PX); % Distribution function for X (Figure 8.10) >> FY = cumsum(PY); % Distribution function for Y >> plot(X,fx,X,FX) % Plotting details omitted

These approximation techniques useful in dealing with functions of random variables, expectations, and conditional expectation and regression.

## 8.3. Problems On Random Vectors and Joint Distributions^{*}

Two cards are selected at random, without replacement, from a standard
deck. Let *X* be the number of aces and *Y* be the number of spades. Under the
usual assumptions, determine the joint distribution and the marginals.

Let *X* be the number of aces and *Y* be the number of spades. Define the events
*A**S*_{i}, *A _{i}*,

*S*, and

_{i}*N*,

_{i}*i*=1,2, of drawing ace of spades, other ace, spade (other than the ace), and neither on the

*i*selection. Let

*P*(

*i*,

*k*)=

*P*(

*X*=

*i*,

*Y*=

*k*).

*P*(2,2)=*P*(*∅*)=0

% type npr08_01 % file npr08_01.m % Solution for Exercise 1. X = 0:2; Y = 0:2; Pn = [132 24 0; 864 144 6; 1260 216 6]; P = Pn/(52*51); disp('Data in Pn, P, X, Y') npr08_01 % Call for mfile Data in Pn, P, X, Y % Result PX = sum(P) PX = 0.8507 0.1448 0.0045 PY = fliplr(sum(P')) PY = 0.5588 0.3824 0.0588

Two positions for campus jobs are open. Two sophomores, three juniors,
and three seniors apply. It is decided to select two at random (each possible pair
equally likely). Let *X* be the number of sophomores and *Y* be the number of
juniors who are selected. Determine the joint distribution for the pair {*X*,*Y*}
and from this determine the marginals for each.

Let *A*_{i},*B*_{i},*C*_{i} be the events of selecting a sophomore, junior, or senior,
respectively, on the *i*th trial. Let *X* be the number of sophomores and *Y* be the
number of juniors selected.

Set

% file npr08_02.m % Solution for Exercise 2. X = 0:2; Y = 0:2; Pn = [6 0 0; 18 12 0; 6 12 2]; P = Pn/56; disp('Data are in X, Y,Pn, P') npr08_02 Data are in X, Y,Pn, P PX = sum(P) PX = 0.5357 0.4286 0.0357 PY = fliplr(sum(P')) PY = 0.3571 0.5357 0.1071

A die is rolled. Let *X* be the number that turns up. A coin is flipped
*X* times. Let *Y* be the number of heads that turn up. Determine the joint distribution
for the pair {*X*,*Y*}. Assume *P*(*X*=*k*)=1/6 for 1≤*k*≤6 and for each
*k*, *P*(*Y*=*j*|*X*=*k*) has the binomial (*k*,1/2) distribution. Arrange the joint
matrix as on the plane, with values of *Y* increasing upward. Determine the marginal
distribution for *Y*. (For a MATLAB based way to determine the joint distribution
see Example 7 from "Conditional Expectation, Regression")

*P*(*X*=*i*,*Y*=*k*)=*P*(*X*=*i*)*P*(*Y*=*k*|*X*=*i*)=(1/6)*P*(*Y*=*k*|*X*=*i*).

% file npr08_03.m % Solution for Exercise 3. X = 1:6; Y = 0:6; P0 = zeros(6,7); % Initialize for i = 1:6 % Calculate rows of Y probabilities P0(i,1:i+1) = (1/6)*ibinom(i,1/2,0:i); end P = rot90(P0); % Rotate to orient as on the plane PY = fliplr(sum(P')); % Reverse to put in normal order disp('Answers are in X, Y, P, PY') npr08_03 % Call for solution m-file Answers are in X, Y, P, PY disp(P) 0 0 0 0 0 0.0026 0 0 0 0 0.0052 0.0156 0 0 0 0.0104 0.0260 0.0391 0 0 0.0208 0.0417 0.0521 0.0521 0 0.0417 0.0625 0.0625 0.0521 0.0391 0.0833 0.0833 0.0625 0.0417 0.0260 0.0156 0.0833 0.0417 0.0208 0.0104 0.0052 0.0026 disp(PY) 0.1641 0.3125 0.2578 0.1667 0.0755 0.0208 0.0026

As a variation of Exercise 3., Suppose a pair of dice is rolled instead
of a single die. Determine the joint distribution for the pair {*X*,*Y*} and from
this determine the marginal distribution for *Y*.

% file npr08_04.m % Solution for Exercise 4. X = 2:12; Y = 0:12; PX = (1/36)*[1 2 3 4 5 6 5 4 3 2 1]; P0 = zeros(11,13); for i = 1:11 P0(i,1:i+2) = PX(i)*ibinom(i+1,1/2,0:i+1); end P = rot90(P0); PY = fliplr(sum(P')); disp('Answers are in X, Y, PY, P') npr08_04 Answers are in X, Y, PY, P disp(P) Columns 1 through 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0005 0 0 0 0 0 0.0013 0.0043 0 0 0 0 0.0022 0.0091 0.0152 0 0 0 0.0035 0.0130 0.0273 0.0304 0 0 0.0052 0.0174 0.0326 0.0456 0.0380 0 0.0069 0.0208 0.0347 0.0434 0.0456 0.0304 0.0069 0.0208 0.0312 0.0347 0.0326 0.0273 0.0152 0.0139 0.0208 0.0208 0.0174 0.0130 0.0091 0.0043 0.0069 0.0069 0.0052 0.0035 0.0022 0.0013 0.0005 Columns 8 through 11 0 0 0 0.0000 0 0 0.0000 0.0001 0 0.0001 0.0003 0.0004 0.0002 0.0008 0.0015 0.0015 0.0020 0.0037 0.0045 0.0034 0.0078 0.0098 0.0090 0.0054 0.0182 0.0171 0.0125 0.0063 0.0273 0.0205 0.0125 0.0054 0.0273 0.0171 0.0090 0.0034 0.0182 0.0098 0.0045 0.0015 0.0078 0.0037 0.0015 0.0004 0.0020 0.0008 0.0003 0.0001 0.0002 0.0001 0.0000 0.0000 disp(PY) Columns 1 through 7 0.0269 0.1025 0.1823 0.2158 0.1954 0.1400 0.0806 Columns 8 through 13 0.0375 0.0140 0.0040 0.0008 0.0001 0.0000

Suppose a pair of dice is rolled. Let *X* be the total number of spots which
turn up. Roll the pair an additional *X* times. Let *Y* be the number of sevens that
are thrown on the *X* rolls. Determine the joint distribution for the pair {*X*,*Y*}
and from this determine the marginal distribution for *Y*. What is the probability
of three or more sevens?

% file npr08_05.m % Data and basic calculations for Exercise 5. PX = (1/36)*[1 2 3 4 5 6 5 4 3 2 1]; X = 2:12; Y = 0:12; P0 = zeros(11,13); for i = 1:11 P0(i,1:i+2) = PX(i)*ibinom(i+1,1/6,0:i+1); end P = rot90(P0); PY = fliplr(sum(P')); disp('Answers are in X, Y, P, PY') npr08_05 Answers are in X, Y, P, PY disp(PY) Columns 1 through 7 0.3072 0.3660 0.2152 0.0828 0.0230 0.0048 0.0008 Columns 8 through 13 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000

The pair has the joint distribution (in m-file npr08_06.m):

Determine the marginal distributions and the corner values for *F*_{XY}. Determine
*P*(*X*+*Y*>2) and *P*(*X*≥*Y*).

npr08_06 Data are in X, Y, P jcalc Enter JOINT PROBABILITIES (as on the plane) P Enter row matrix of VALUES of X X Enter row matrix of VALUES of Y Y Use array operations on matrices X, Y, PX, PY, t, u, and P disp([X;PX]') -2.3000 0.2300 -0.7000 0.1700 1.1000 0.2000 3.9000 0.2020 5.1000 0.1980 disp([Y;PY]') 1.3000 0.2980 2.5000 0.3020 4.1000 0.1900 5.3000 0.2100 jddbn Enter joint probability matrix (as on the plane) P To view joint distribution function, call for FXY disp(FXY) 0.2300 0.4000 0.6000 0.8020 1.0000 0.1817 0.3160 0.4740 0.6361 0.7900 0.1380 0.2400 0.3600 0.4860 0.6000 0.0667 0.1160 0.1740 0.2391 0.2980 P1 = total((t+u>2).*P) P1 = 0.7163 P2 = total((t>=u).*P) P2 = 0.2799

The pair has the joint distribution (in m-file npr08_07.m):

t = | -3.1 | -0.5 | 1.2 | 2.4 | 3.7 | 4.9 |

u = 7.5 | 0.0090 | 0.0396 | 0.0594 | 0.0216 | 0.0440 | 0.0203 |

4.1 | 0.0495 | 0 | 0.1089 | 0.0528 | 0.0363 | 0.0231 |

-2.0 | 0.0405 | 0.1320 | 0.0891 | 0.0324 | 0.0297 | 0.0189 |

-3.8 | 0.0510 | 0.0484 | 0.0726 | 0.0132 | 0 | 0.0077 |

Determine the marginal distributions and the corner values for *F*_{XY}. Determine
*P*(1≤*X*≤4,*Y*>4) and *P*(|*X*–*Y*|≤2).

npr08_07 Data are in X, Y, P jcalc Enter JOINT PROBABILITIES (as on the plane) P Enter row matrix of VALUES of X X Enter row matrix of VALUES of Y Y Use array operations on matrices X, Y, PX, PY, t, u, and P disp([X;PX]') -3.1000 0.1500 -0.5000 0.2200 1.2000 0.3300 2.4000 0.1200 3.7000 0.1100 4.9000 0.0700 disp([Y;PY]') -3.8000 0.1929 -2.0000 0.3426 4.1000 0.2706 7.5000 0.1939 jddbn Enter joint probability matrix (as on the plane) P To view joint distribution function, call for FXY disp(FXY) 0.1500 0.3700 0.7000 0.8200 0.9300 1.0000 0.1410 0.3214 0.5920 0.6904 0.7564 0.8061 0.0915 0.2719 0.4336 0.4792 0.5089 0.5355 0.0510 0.0994 0.1720 0.1852 0.1852 0.1929 M = (1<=t)&(t<=4)&(u>4); P1 = total(M.*P) P1 = 0.3230 P2 = total((abs(t-u)<=2).*P) P2 = 0.3357

The pair has the joint distribution (in m-file npr08_08.m):

t = | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 |

u = 12 | 0.0156 | 0.0191 | 0.0081 | 0.0035 | 0.0091 | 0.0070 | 0.0098 | 0.0056 | 0.0091 | 0.0049 |

10 | 0.0064 | 0.0204 | 0.0108 | 0.0040 | 0.0054 | 0.0080 | 0.0112 | 0.0064 | 0.0104 | 0.0056 |

9 | 0.0196 | 0.0256 | 0.0126 | 0.0060 | 0.0156 | 0.0120 | 0.0168 | 0.0096 | 0.0056 | 0.0084 |

5 | 0.0112 | 0.0182 | 0.0108 | 0.0070 | 0.0182 | 0.0140 | 0.0196 | 0.0012 | 0.0182 | 0.0038 |

3 | 0.0060 | 0.0260 | 0.0162 | 0.0050 | 0.0160 | 0.0200 | 0.0280 | 0.0060 | 0.0160 | 0.0040 |

-1 | 0.0096 | 0.0056 | 0.0072 | 0.0060 | 0.0256 | 0.0120 | 0.0268 | 0.0096 | 0.0256 | 0.0084 |

-3 | 0.0044 | 0.0134 | 0.0180 | 0.0140 | 0.0234 | 0.0180 | 0.0252 | 0.0244 | 0.0234 | 0.0126 |

-5 | 0.0072 | 0.0017 | 0.0063 | 0.0045 | 0.0167 | 0.0090 | 0.0026 | 0.0172 | 0.0217 | 0.0223 |

Determine the marginal distributions. Determine *F*_{XY}(10,6) and *P*(*X*>*Y*).

npr08_08 Data are in X, Y, P jcalc - - - - - - - - - Use array operations on matrices X, Y, PX, PY, t, u, and P disp([X;PX]') 1.0000 0.0800 3.0000 0.1300 5.0000 0.0900 7.0000 0.0500 9.0000 0.1300 11.0000 0.1000 13.0000 0.1400 15.0000 0.0800 17.0000 0.1300 19.0000 0.0700 disp([Y;PY]') -5.0000 0.1092 -3.0000 0.1768 -1.0000 0.1364 3.0000 0.1432 5.0000 0.1222 9.0000 0.1318 10.0000 0.0886 12.0000 0.0918 F = total(((t<=10)&(u<=6)).*P) F = 0.2982 P = total((t>u).*P) P = 0.7390

Data were kept on the effect of training time on the time to perform
a job on a production line. *X* is the amount of training, in hours, and *Y* is
the time to perform the task, in minutes. The data are as follows (in m-file npr08_09.m):

t = | 1 | 1.5 | 2 | 2.5 | 3 |

u = 5 | 0.039 | 0.011 | 0.005 | 0.001 | 0.001 |

4 | 0.065 | 0.070 | 0.050 | 0.015 | 0.010 |

3 | 0.031 | 0.061 | 0.137 | 0.051 | 0.033 |

2 | 0.012 | 0.049 | 0.163 | 0.058 | 0.039 |

1 | 0.003 | 0.009 | 0.045 | 0.025 | 0.017 |

Determine the marginal distributions. Determine *F*_{XY}(2,3) and
*P*(*Y*/*X*≥1.25).

npr08_09 Data are in X, Y, P jcalc - - - - - - - - - - - - Use array operations on matrices X, Y, PX, PY, t, u, and P disp([X;PX]') 1.0000 0.1500 1.5000 0.2000 2.0000 0.4000 2.5000 0.1500 3.0000 0.1000 disp([Y;PY]') 1.0000 0.0990 2.0000 0.3210 3.0000 0.3130 4.0000 0.2100 5.0000 0.0570 F = total(((t<=2)&(u<=3)).*P) F = 0.5100 P = total((u./t>=1.25).*P) P = 0.5570

**For the joint densities in Exercises 10-22 below**

Sketch the region of definition and determine analytically the marginal density functions

*f*and_{X}*f*._{Y}Use a discrete approximation to plot the marginal density

*f*and the marginal distribution function_{X}*F*._{X}Calculate analytically the indicated probabilities.

Determine by discrete approximation the indicated probabilities.

*f*_{XY}(*t*,*u*)=1 for 0≤*t*≤1, 0≤*u*≤2(1–*t*).

Region is triangle with vertices (0,0), (1,0), (0,2).

tuappr Enter matrix [a b] of X-range endpoints [0 1] Enter matrix [c d] of Y-range endpoints [0 2] Enter number of X approximation points 200 Enter number of Y approximation points 400 Enter expression for joint density (t<=1)&(u<=2*(1-t)) Use array operations on X, Y, PX, PY, t, u, and P fx = PX/dx; FX = cumsum(PX); plot(X,fx,X,FX) % Figure not reproduced M1 = (t>0.5)&(u>1); P1 = total(M1.*P) P1 = 0 % Theoretical = 0 M2 = (t<=0.5)&(u>0.5); P2 = total(M2.*P) P2 = 0.5000 % Theoretical = 1/2 P3 = total((u<=t).*P) P3 = 0.3350 % Theoretical = 1/3

*f*_{XY}(*t*,*u*)=1/2 on the square with vertices at
.

The region is bounded by the lines *u*=1+*t*, *u*=1–*t*, *u*=3–*t*,
and *u*=*t*–1

tuappr Enter matrix [a b] of X-range endpoints [0 2] Enter matrix [c d] of Y-range endpoints [0 2] Enter number of X approximation points 200 Enter number of Y approximation points 200 Enter expression for joint density 0.5*(u<=min(1+t,3-t))& ... (u>=max(1-t,t-1)) Use array operations on X, Y, PX, PY, t, u, and P fx = PX/dx; FX = cumsum(PX); plot(X,fx,X,FX) % Plot not shown M1 = (t>1)&(u>1); PM1 = total(M1.*P) PM1 = 0.2501 % Theoretical = 1/4 M2 = (t<=1/2)&(u>1); PM2 = total(M2.*P) PM2 = 0.0631 % Theoretical = 1/16 = 0.0625 M3 = u<=t; PM3 = total(M3.*P) PM3 = 0.5023 % Theoretical = 1/2

*f*_{XY}(*t*,*u*)=4*t*(1–*u*) for 0≤*t*≤1, 0≤*u*≤1.

Region is the unit square.

*P*3 = ∫

_{0}

^{1}∫

_{0}

^{t}4

*t*( 1 –

*u*)

*d*

*u*

*d*

*t*= 5 / 6

tuappr Enter matrix [a b] of X-range endpoints [0 1] Enter matrix [c d] of Y-range endpoints [0 1] Enter number of X approximation points 200 Enter number of Y approximation points 200 Enter expression for joint density 4*t.*(1 - u) Use array operations on X, Y, PX, PY, t, u, and P fx = PX/dx; FX = cumsum(PX); plot(X,fx,X,FX) % Plot not shown M1 = (1/2<t)&(t<3/4)&(u>1/2); P1 = total(M1.*P) P1 = 0.0781 % Theoretical = 5/64 = 0.0781 M2 = (t<=1/2)&(u>1/2); P2 = total(M2.*P) P2 = 0.0625 % Theoretical = 1/16 = 0.0625 M3 = (u<=t); P3 = total(M3.*P) P3 = 0.8350 % Theoretical = 5/6 = 0.8333

for 0≤*t*≤2,
0≤*u*≤2.

Region is the square .

*P*3 = ∫

_{0}

^{2}∫

_{0}

^{t}(

*t*+

*u*)

*d*

*u*

*d*

*t*= 1 / 2

tuappr Enter matrix [a b] of X-range endpoints [0 2] Enter matrix [c d] of Y-range endpoints [0 2] Enter number of X approximation points 200 Enter number of Y approximation points 200 Enter expression for joint density (1/8)*(t+u) Use array operations on X, Y, PX, PY, t, u, and P fx = PX/dx; FX = cumsum(PX); plot(X,fx,X,FX) M1 = (t>1/2)&(u>1/2); P1 = total(M1.*P) P1 = 0.7031 % Theoretical = 45/64 = 0.7031 M2 = (t<=1)&(u>1); P2 = total(M2.*P) P2 = 0.2500 % Theoretical = 1/4 M3 = u<=t; P3 = total(M3.*P) P3 = 0.5025 % Theoretical = 1/2

*f*_{XY}(*t*,*u*)=4*u**e*^{–2t} for

Region is strip bounded by

tuappr Enter matrix [a b] of X-range endpoints [0 3] Enter matrix [c d] of Y-range endpoints [0 1] Enter number of X approximation points 400 Enter number of Y approximation points 200 Enter expression for joint density 4*u.*exp(-2*t) Use array operations on X, Y, PX, PY, t, u, and P M2 = (t > 0.5)&(u > 0.5)&(u<3/4); p2 = total(M2.*P) p2 = 0.1139 % Theoretical = (5/16)exp(-1) = 0.1150 p3 = total((t<u).*P) p3 = 0.7047 % Theoretical = 0.7030

for
0≤*t*≤2, 0≤*u*≤1+*t*.

Region bounded by

*F*

_{XY}( 1 , 1 ) = ∫

_{0}

^{1}∫

_{0}

^{1}

*f*

_{XY}(

*t*,

*u*)

*d*

*u*

*d*

*t*= 3 / 44

tuappr Enter matrix [a b] of X-range endpoints [0 2] Enter matrix [c d] of Y-range endpoints [0 3] Enter number of X approximation points 200 Enter number of Y approximation points 300 Enter expression for joint density (3/88)*(2*t+3*u.^2).*(u<=1+t) Use array operations on X, Y, PX, PY, t, u, and P fx = PX/dx; FX = cumsum(PX); plot(X,fx,X,FX) MF = (t<=1)&(u<=1); F = total(MF.*P) F = 0.0681 % Theoretical = 3/44 = 0.0682 M1 = (t<=1)&(u>1); P1 = total(M1.*P) P1 = 0.1172 % Theoretical = 41/352 = 0.1165 M2 = abs(t-u)<1; P2 = total(M2.*P) P2 = 0.9297 % Theoretical = 329/352 = 0.9347

*f*_{XY}(*t*,*u*)=12*t*^{2}*u* on the parallelogram with vertices

Region bounded by

*P*3 = 1 –

*P*2 = 13 / 16

tuappr Enter matrix [a b] of X-range endpoints [-1 1] Enter matrix [c d] of Y-range endpoints [0 1] Enter number of X approximation points 400 Enter number of Y approximation points 200 Enter expression for joint density 12*u.*t.^2.*((u<=t+1)&(u>=t)) Use array operations on X, Y, PX, PY, t, u, and P p1 = total((t<=1/2).*P) p1 = 0.4098 % Theoretical = 33/80 = 0.4125 M2 = (t<1/2)&(u<=1/2); p2 = total(M2.*P) p2 = 0.1856 % Theoretical = 3/16 = 0.1875 P3 = total((u>=1/2).*P) P3 = 0.8144 % Theoretical = 13/16 = 0.8125

for 0≤*t*≤2,
0≤*u*≤min{1,2–*t*}

Region is bounded by

tuappr Enter matrix [a b] of X-range endpoints [0 2] Enter matrix [c d] of Y-range endpoints [0 1] Enter number of X approximation points 400 Enter number of Y approximation points 200 Enter expression for joint density (24/11)*t.*u.*(u<=2-t) Use array operations on X, Y, PX, PY, t, u, and P M1 = (t<=1)&(u<=1); P1 = total(M1.*P) P1 = 0.5447 % Theoretical = 6/11 = 0.5455 P2 = total((t>1).*P) P2 = 0.4553 % Theoretical = 5/11 = 0.4545 P3 = total((t<u).*P) P3 = 0.2705 % Theoretical = 3/11 = 0.2727

for 0≤*t*≤2,
0≤*u*≤max{2–*t*,*t*}

Region is bounded by

tuappr Enter matrix [a b] of X-range endpoints [0 2] Enter matrix [c d] of Y-range endpoints [0 2] Enter number of X approximation points 200 Enter number of Y approximation points 200 Enter expression for joint density (3/23)*(t+2*u).*(u<=max(2-t,t)) Use array operations on X, Y, PX, PY, t, u, and P M1 = (t>=1)&(u>=1); P1 = total(M1.*P) P1 = 0.2841 13/46 % Theoretical = 13/46 = 0.2826 P2 = total((u<=1).*P) P2 = 0.5190 % Theoretical = 12/23 = 0.5217 P3 = total((u<=t).*P) P3 = 0.6959 % Theoretical = 16/23 = 0.6957

, for
0≤*t*≤2, 0≤*u*≤min{2,3–*t*}

*P*(

*X*≥ 1 ,

*Y*≥ 1 ) ,

*P*(

*X*≤ 1 ,

*Y*≤ 1 ) ,

*P*(

*Y*<

*X*)

Region has two parts: (1) 0≤*t*≤1,0≤*u*≤2 (2) 1<*t*≤2,0≤*u*≤3–*t*

tuappr Enter matrix [a b] of X-range endpoints [0 2] Enter matrix [c d] of Y-range endpoints [0 2] Enter number of X approximation points 200 Enter number of Y approximation points 200 Enter expression for joint density (12/179)*(3*t.^2+u).* ... (u<=min(2,3-t)) Use array operations on X, Y, PX, PY, t, u, and P fx = PX/dx; FX = cumsum(PX); plot(X,fx,X,FX) M1 = (t>=1)&(u>=1); P1 = total(M1.*P) P1 = 2312 % Theoretical = 41/179 = 0.2291 M2 = (t<=1)&(u<=1); P2 = total(M2.*P) P2 = 0.1003 % Theoretical = 18/179 = 0.1006 M3 = u<=min(t,3-t); P3 = total(M3.*P) P3 = 0.7003 % Theoretical = 1001/1432 = 0.6990

for
0≤*t*≤2, 0≤*u*≤min{1+*t*,2}

Region is in two parts:

(2)

*f*

_{X}(

*t*) =

*I*

_{ [ 0 , 1 ] }(

*t*) ∫

_{0}

^{ 1 + t }

*f*

_{XY}(

*t*,

*u*)

*d*

*u*+

*I*

_{ ( 1 , 2 ] }(

*t*) ∫

_{0}

^{2}

*f*

_{XY}(

*t*,

*u*)

*d*

*u*=

*f*

_{Y}(

*u*) =

*I*

_{ [ 0 , 1 ] }(

*u*) ∫

_{0}

^{2}

*f*

_{XY}(

*t*,

*u*)

*d*

*t*+

*I*

_{ ( 1 , 2 ] }(

*u*) ∫

^{2}

_{ u – 1 }

*f*

_{XY}(

*t*,

*u*)

*d*

*t*=

tuappr Enter matrix [a b] of X-range endpoints [0 2] Enter matrix [c d] of Y-range endpoints [0 2] Enter number of X approximation points 200 Enter number of Y approximation points 200 Enter expression for joint density (12/227)*(3*t+2*t.*u).* ... (u<=min(1+t,2)) Use array operations on X, Y, PX, PY, t, u, and P M1 = (t<=1/2)&(u<=3/2); P1 = total(M1.*P) P1 = 0.0384 % Theoretical = 139/3632 = 0.0383 M2 = (t<=3/2)&(u>1); P2 = total(M2.*P) P2 = 0.3001 % Theoretical = 68/227 = 0.2996 M3 = u<t; P3 = total(M3.*P) P3 = 0.6308 % Theoretical = 144/227 = 0.6344

for
0≤*t*≤2, 0≤*u*≤min{2*t*,3–*t*}

Region bounded by

*P*3 = ∫

_{0}

^{2}∫

_{0}

^{ t / 2 }(

*t*+ 2

*u*)

*d*

*u*

*d*

*t*= 4 / 13

tuappr Enter matrix [a b] of X-range endpoints [0 2] Enter matrix [c d] of Y-range endpoints [0 2] Enter number of X approximation points 400 Enter number of Y approximation points 400 Enter expression for joint density (2/13)*(t+2*u).*(u<=min(2*t,3-t)) Use array operations on X, Y, PX, PY, t, u, and P P1 = total((t<1).*P) P1 = 0.3076 % Theoretical = 4/13 = 0.3077 M2 = (t>=1)&(u<=1); P2 = total(M2.*P) P2 = 0.3844 % Theoretical = 5/13 = 0.3846 P3 = total((u<=t/2).*P) P3 = 0.3076 % Theoretical = 4/13 = 0.3077

for 0≤*u*≤1.

*P*( 1 / 2 ≤

*X*≤ 3 / 2 ,

*Y*≤ 1 / 2 )

Region is rectangle bounded by

tuappr Enter matrix [a b] of X-range endpoints [0 2] Enter matrix [c d] of Y-range endpoints [0 1] Enter number of X approximation points 400 Enter number of Y approximation points 200 Enter expression for joint density (3/8)*(t.^2+2*u).*(t<=1) ... + (9/14)*(t.^2.*u.^2).*(t > 1) Use array operations on X, Y, PX, PY, t, u, and P M = (1/2<=t)&(t<=3/2)&(u<=1/2); P = total(M.*P) P = 0.1228 % Theoretical = 55/448 = 0.1228