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# 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 if it is a success, if it is a failure.
• A consecutive 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. 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,

• Note that the "location" of the s don't matter. It is the count of 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.

2. Binomial distribution.
• The number of successes in Bernoulli trials is a random variable, .
• What is the probability that out of trials are success?

• There are

• So the total probability of successes out of trials is given by

• Mean and variance: consider the binomial distribution as the sum of independent Bernoulli trials. Thus

• 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

• The mean is calculated as follows.

because the sequence converges, so we can exchange the order of summation and the differentiation.

• Variance

• Example 6.11 on page 199

4. Poisson distribution. The Poisson distribution has the following pdf

• Poisson distribution property: mean and variance are the same

• The cdf of the Poisson distribution

because is a constant and the increase rate of is much faster than that of , 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

Next: Contineous Distributions Up: Statistical Model of Simulation Previous: Useful Statistical Models
Meng Xiannong 2002-10-18