# 1: Continuous Distributions (Worksheet)

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Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.

A continuous probability distribution represents the histogram of sampling a random variable $$x$$ that can take on any value (i.e., is continuous). There are many different characteristics used to describe a distribution $$D(x)$$. Each of the characteristics below commonly show up in discussions of quantum mechanics.

• The integrated value ($$A$$): This is the sum over the distribution over all possible values of $$x$$. Graphically, this is the area under the distribution. $A= \int_{\infty}^{\infty} D(x)\, dx \label{1}$ If the range of $$x$$ is less than $$-\infty$$ to $$\infty$$ (e.g., from $$a$$ to $$b)$$, then $$A$$ is given by $A = \int_{a}^{b} D(x)\, dx \label{2}$
• The expectation value ($$\langle x \rangle$$): This is a different term that is used synonymously with the average (or mean) of $$x$$ over the distribution. The mean is given by $\bar{x} = \langle x \rangle = \int_{a}^{b} x D(x)\, dx \label{3}$ Notice the difference between Equations $$\ref{2}$$ and $$\ref{3}$$.
• The most probable value ($$x_{mp}$$): This is the value of $$x$$ at the peak of $$D(x)$$. This is determined from basic calculus for determining extrema and via identifying when the derivative of the distribution is zero. $\left( \dfrac{dD(x)}{dx} \right)_{x_{mp}} = 0 \label{4}$
• The standard deviation ($$\sigma_x$$): This is the a measure of the spread of the distribution and is given by $\sigma_x^2 = \int_{a}^{b} (x-\bar{x})^2 D(x) \,dx \label{5}$

## Applications

One example distribution is the Sine-squared distribution:

$D(x) = N_o \sin^2 (N_1 x)$

where $$N_o$$ and $$n_1$$ are constants. The range of $$x$$ goes from $$0$$ to $$\pi$$.

Another is the Gaussian distribution:

$D(x) = N_oE^{-N_1x^2}$

where $$N_o$$ and $$N_1$$ are constants. The range of $$x$$ goes from $$-\infty$$ to $$\infty$$.

### Q1

Plot the Sine-squared and Gaussian distributions.

### Q2

Calculate the four characteristics defined above for the Gaussian distribution.

## Interpretation of a Probability Distribution

A probability distribution is defined such that the likelihood of a value of $$x$$ being sampled between $$a$$ and $$b$$ is equal to the integral (area under the curve) between $$a$$ and $$b$$, e.g., Equation $$\ref{2}$$. This integrated values is always positive and if the full range of possible $$x$$ values are integrated over, then the area under the curve from negative infinity to positive infinity is one.

$A = \int_{a}^{b} D(x) dx = 1 \label{6}$

For continuous probability distributions, we cannot calculate exact probability for a specific outcome, but instead we calculate a probability for a range of outcomes (e.g., the probability that a sampled value of $$x$$ is greater than 10). The probability that a continuous variable will take a specific value is equal to zero. Because of this, we can never express continuous probability distribution in a tabular form. Another way to say this is that the probability for $$x$$ to take any single value $$a$$ (that is $$a ≤ x ≤ a$$) is zero, because an integral with the same upper and lower limits is always equal to zero.

$A = \int_{a}^{a} D(x) dx = 0 \label{7}$

### Q3

What is the expression for finding $$x$$ exactly at 0 for a Gaussian probability distribution?

### Q4

What is the expression for finding $$x$$ between one standard deviation on each side of 0 for a Gaussian probability distribution?

### Q5

Use the information above to identify the constants $$N_o$$ and $$N_1$$ in the Gaussian Distribution? These integrals may be needed:

$\int _{-\infty }^{\infty }e^{-a(x+b)^{2}}\,dx={\sqrt {\frac {\pi }{a}}}.$

$\int _{-\infty }^{\infty }e^{-x^{2}}dx=2\int _{0}^{\infty }e^{-x^{2}}dx$

### Q6

What is the expression for finding $$x$$ between $$0$$ and $$+\infty$$ for a Gaussian probability distribution when $$N_o$$ and $$N_1$$ are determined?

### Q7

The Gaussian distribution is called a Normal distribution (sometimes called a bell curve) if its standard deviation is 1 and the area under the distribution is 1 from $$-\infty$$ to $$+\infty$$. What is the mathematical formula of the Normal distribution.

### Q8

A function converges if it approach a limit more and more closely as an argument (variable) of the function increases or decreases (i.e., Horizontal Asymptotes). For example, the function $$y = \frac{1}{x}$$ converges to zero as $$x$$ increases. Gaussian functions converge to zero with respect to the argument approaching $$-\infty$$ and $$+\infty$$. Explain why this is a required property of a probability distribution.