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.