Discrete Random Variables

Here we are going to study a few discrete random variable distributions.

1. Bernoulli trials and the Bernoulli distribution.
   • A Bernoulli trail is an experiment with result of success or failure.
   • We can use a random variable to model this phenomena. Let $X_j = 1$ if it is a success, $X_j = 0$ if it is a failure.
   • A consecutive $n$ Bernoulli trails are called a Bernoulli process if
     • the trails are independent of each other;
     • each trail has only two possible outcomes (success or failure, true or false, etc.); and
     • the probability of a success remains constant
   • The following relations hold.
     1. $p(x_1, x_2, ..., x_n) = p_1(x_1) * p_2(x_2) * ... * p_n(x_n)$ which means the probability of the result of a sequence of events is equal to the product of the probabilities of each event.
     2. Because the events are independent and the probability remains the same,
        
        $$p_j(x_j) = p (x_j) = \left\{ \begin{array}{ll} p & x_j = 1, ... ...,n \\ 1-p = q & x_j = 0, j = 1,2,...,n \end{array} \right.$$
        
        

   • Note that the "location" of the $p_i$s don't matter. It is the count of $p_i$s that is important.
   • Examples of Bernoulli trails include: a conscutive throwing of a "fair" coin, counting heads and tails; a pass or fail test on a sequence of a particular components of the "same" quality; and others of the similar type.
   • For one trial, the distribution above is called the Bernoulli distribution. The mean and the variance is as follows.
     
     $$E(X_j) = 0 * q + 1 * p = p$$
     
     
     
     
     $$V(X_j) = E[X^2] - (E[X])^2 = [0^2*q + 1^2 * p] - p^2 = p (1-p)$$
     
     

2. Binomial distribution.
   • The number of successes in $n$ Bernoulli trials is a random variable, $X$.
   • What is the probability that $m$ out of $n$ trials are success?
     
     $$p_{m, n} = p^m q^{n-m}$$
     
     

   • There are
     
     $$\left( \begin{array}{l} n \\ x \\ \end{array} \right) = \frac{n!}{x!(n-x)!}$$
     
     

   • So the total probability of $m$ successes out of $n$ trials is given by
     
     $$p(x) = \left\{ \begin{array}{ll} \left( \begin{array}{l} n ... ... 0,1,2,...,n \\ 0 & {\rm otherwise} \\ \end{array} \right.$$
     
     

   • Mean and variance: consider the binomial distribution as the sum of $n$ independent Bernoulli trials. Thus
     
     $$E[X] = p + p + ... + p = np$$
     
     
     
     
     $$V(X) = pq + pq + ... + pq = npq$$
     
     

   • Example 6.10 on page 198

3. Geometric distribution.
   • The number of trials needed in a Bernoulli trial to achieve the first success is a random variable that follows the geometric distribution.
   • The distribution is given by
     
     $$p(x) = \left\{ \begin{array}{ll} q^{x-1} p & x = 1,2,... \\ 0 & {\rm otherwise} \\ \end{array} \right.$$
     
     

   • The mean is calculated as follows.
     
     $$E(X) = \sum_{x=1}^\infty x q^{x-1} p = \sum_{x=1}^\infty xq^{x-1}p$$
     
     
     
     
     $$E(X) = p \sum_{x=1}^\infty \frac{d}{dq} q^x$$
     
     
     because the sequence converges, so we can exchange the order of summation and the differentiation.
     
     $$E(X) = p \frac{d}{dx} \sum_{x=0}^\infty q^x = p \frac{d}{dq} \frac{1}{1-q}$$
     
     
     
     
     $$= p \frac{1}{(1-q)^2} = \frac{1}{p}$$
     
     

   • Variance $V(X) = \frac{q}{p^2}$

   • Example 6.11 on page 199

4. Poisson distribution. The Poisson distribution has the following pdf
   
   $$p(x) = \left\{ \begin{array}{ll} \frac{e^{-\alpha} \alpha^x}... ...x = 0,1,2,... \\ 0 & {\rm otherwise}\\ \end{array} \right.$$
   
   
   • Poisson distribution property: mean and variance are the same
     
     $$E(X) = \alpha = V(X)$$
     
     

   • The cdf of the Poisson distribution
     
     $$F(x) = \sum_{i=0}^x \frac{e^{-\alpha} \alpha^i}{i!}$$
     
     

     because $\alpha$ is a constant and the increase rate of $i!$ is much faster than that of $\alpha^i$, the distribution is mostly determined by the first a few items.

   • Poisson distribution is most useful in Poisson process which is used most frequently in describing such phenomena as telephone call arrivals.
   • Example 6.12, 6.13, 6.14 on page 200, page 201