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.