Lecture 10: Some Important Examples of Continuous Distributions

Some important continuous distributions. We shall give interpretations of these distribution as time goes by. For now, it will be a good review of calculus and algebra to consider the following simply as formulae to be manipulated. We shall only give the value of the probability density function for the range values of the random variable. At any other value it is .

Continuous uniform:

A random variable with range [a, b] is said to have the uniform distribution if

$$f_X(y) = \frac{1}{b-a}\;\;{\rm\;for}\;\;y \in [a,b]$$

The mean and variance of such a random variable are given by

$$E[X] = \frac{a + b}{2} \;\;\;\;\;\;\;\;\;\;\;\; V[X] = \frac{(b-a)^2}{12}$$

Exponential:

Pick a positive real number $\lambda$. A random variable with range $(0,\infty)$ is said to have a exponential distribution if

$$f_X(y) = \lambda e^{-\lambda y} {\rm\;for}\;\;y \in (0,\infty)$$

The mean and variance of such a random variable are given by

$$E[X] = 1/\lambda \;\;\;\;\;\;\;\;\;\;\;\; V[X] = 1/\lambda^2$$

Gamma:

Recall from calculus that the gamma function, $\Gamma(t)$ is defined by the formula

$$\Gamma(t) = \int_0^\infty u^{t-1}e^{-u}du, t \gt 0$$

Integration by part shows that $\Gamma(t+1) = t\Gamma(t)$. If t is an integer, $\Gamma(t+1) = t!$, and $\Gamma(1/2) = \sqrt{\pi}$.

Now, pick a two positive real numbers, $\alpha$ and $\beta$. A random variable with range $(0,\infty)$ is said to have a gamma-type distribution if

$$f_X(y) = \frac{y^{\alpha-1}e^{-y/\beta}}{\beta^\alpha\Gamma(\alpha)} {\rm\;for}\;\;y \in (0,\infty)$$

The mean and variance of such a random variable are given by

$$E[X] = \beta\alpha \;\;\;\;\;\;\;\;\;\;\;\; V[X] = \beta^2\alpha$$

Chi-square:

Pick a positive integer v. A random variable with range $(0,\infty)$ is said to have a chi-square distribution with v degrees of freedom if

$$f_X(y) = \frac{y^{(v/2)-1}e^{-y/2}}{2^{(v/2)}\Gamma(v/2)} {\rm\;for}\;\;y \in (0,\infty)$$

The mean and variance of such a random variable are given by

$$E[X] = v \;\;\;\;\;\;\;\;\;\;\;\; V[X] = 2v$$

Note that the exponential and the chi-square are special cases of the gamma.

Normal:

Pick a real number $\mu$ and a positive real number $\sigma$. A random variable with range $(-\infty,\infty)$ is said to have a normal distribution if

$$f_X(y) = \frac{1}{\sigma\sqrt{2\pi}}\exp(-\frac{(y-\mu)^2}{2\sigma^2}) {\rm\;for}\;\;y \in (-\infty,\infty)$$

The mean and variance of such a random variable are given by

$$E[X] = \mu \;\;\;\;\;\;\;\;\;\;\;\; V[X] = \sigma^2$$

Beta:

Pick two positive real numbers $\alpha$ and $\beta$.Recall from calculus that the beta function, $B(\alpha,\beta)$ is defined by

$$B(\alpha,\beta) = \int_0^1u^{\alpha-1}(1-u)^{\beta-1}du$$

With a little ingenuity it can be shown that

$$B(\alpha,\beta) = \frac{\Gamma(\alpha)\cdot\Gamma(\beta)}{\Gamma(\alpha+\beta)}$$

A random variable with range (0,1) is said to have a beta distribution if

$$f_X(y) = \frac{1}{B(\alpha,\beta)}y^{\alpha-1}(1-y)^{\beta - 1} {\rm\;for}\;\;y \in (0,1)$$

The mean and variance of such a random variable are given by

$$E[X] = \frac{\alpha}{\alpha+\beta} \;\;\;\;\;\;\;\;\;\;\;\; V[X] = \frac{\alpha\cdot\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$$