Lecture 7: Some Important Examples of Discrete Random Variables

Some important discrete distributions. We shall give interpretations of these distribution in terms of coin tossing, etc. 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 mass function for the range values of the random variable. At any other value it is 0.

Uniform:

Suppose N is a positive integer and k is any integer. A random variable R whose range is $\{N+1,\dots,N+k\}$is said to have the uniform distribution if

$$p_R(y) = \frac{1}{N}\;\;{\rm\;for}\;\;y = k+1,\dots,k+N$$

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

$$E[R] = k + \frac{N+1}{2} \;\;\;\;\;\;\;\;\;\;\;\; V[R] = \frac{N^2-1}{12}$$

Such a random variable might arise from the experiment, choose an integer at random from the integers between N+1 and N+k.

Binomial:

Pick a real number $p\in [0,1]$. A random variable with range $\{0,1,\dots,N\}$ is said to have a binomial distribution if

$$p_R(y) = \frac{N!}{y!(N-y)!}p^y(1-p)^{N-y}{\rm\;for}\;\;y = 0,1,\dots,N$$

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

$$E[R] = N\cdot p \;\;\;\;\;\;\;\;\;\;\;\; V[R] = N\cdot p(1-p)$$

Such a random variable might arise from counting the number of heads in N independent tosses of a coin with probablity p of heads on a single toss.

Geometric:

Pick a real number $p\in (0,1]$. A random variable with range $\{1, 2,\dots,\}$ is said to have a geometric distribution if

$$p_R(y) = (1-p)^{y-1}p{\rm\;for}\;\;y = 1, 2, \dots$$

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

$$E[R] = \frac{1}{p} \;\;\;\;\;\;\;\;\;\;\;\; V[R] = \frac{1-p}{p^2}$$

Such a random variable might arise from counting the number of independent tosses of a coin with probility heads until the first head is obtained.

Negative Binomial:

Pick a real number $p\in [0,1]$ and a positive integer r . A random variable with range $\{r, r+1,\dots,\}$ is said to have a negative binomial distribution if

$$p_R(y) = \frac{(y-1)!}{(r-1)!(y-r)!}p^{r}(1-p)^{y-r} {\rm\;for}\;\;y = r, r+1, \dots$$

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

$$E[R] = \frac{r}{p} \;\;\;\;\;\;\;\;\;\;\;\; V[R] = \frac{r(1-p)}{p^2}$$

This is the generalization of the previous example to the number of tosses for to obtain r heads.

Hypergeometric:

Pick three positive integers N and n and r with $n \leq N$ and r < N. A random variable with range $\{0, 1,\dots,\min\{n,r\}\;\}$ is said to have a hypergeometric distribution if

$$p_R(y) = \frac{\mbox{$\left(\begin{array} {c}r\ y\end{array... ...\;for}\;\;y = 0, 1, \dots, \min\{n,r\}\;{\rm and\;} n-y\leq N-r$$

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

$$E[R] = \frac{n\cdot r}{N} \;\;\;\;\;\;\;\;\;\;\;\; V[R] = n\frac{r}{N}\cdot\frac{N-r}{N}\cdot\frac{N-n}{N-1}$$

Suppose an urn contains N balls, r of which are red and N-r of which are green. n balls are selected without replacement. R represents the number of red balls.

Poisson:

Pick a positive real number $\lambda$. A random variable with range $\{0, 1,\dots\}$ is said to have a Poisson distribution if

$$p_R(y) = \frac{\lambda^y}{y!}e^{-\lambda}\;{\rm\;for}\;\;y = 0, 1, \dots$$

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

$$E[R] = \lambda \;\;\;\;\;\;\;\;\;\;\;\; V[R] = \lambda$$

This a little more complicated to explain, and we shall leave this to another time.

Homework, Due Wednesday, September 30, 1998

1.

Derive the formulae for the mean, variance and standard deviation for the discrete uniform distribution, for the binomial distribution, for the geometric distribution and for the Poisson distribution. You will find it helpful to read the proofs of Theorems 3.7, 3.8, and 3.11 in your text.

2.

Graph the probability mass function and the distribution function for a random variable with a binomial distribution with expected value of 4 and variance of 2.

3.

Graph the probability mass function and the distribution function for a random variable with a binomial distribution with expected value of 3 and variance of 2.

4.

Graph the probability mass function and the distribution function for a random variable with a hypergeomtric distribution with N = 26, n = 13 and r = 5.

5.

Graph the probability mass function and the distribution function for a random variable with a hypergeomtric distribution with N = 26, n = 13 and r = 6.

6.

Suppose that the random variable R has a geometric distribution, and H(z) = e^tz. Compute E[H(R)]. Be careful to indicate for which values of t E[H(R)] is defined.

7.

Suppose that the random variable R has a Poisson distribution, and H(z) = e^tz. Compute E[H(R)]. Be careful to indicate for which values of t E[H(R)] is defined.