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

*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.
- which means the probability of the result of a sequence of events is equal to the product of the probabilities of each event.
- 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.

*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

*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

- The number of trials needed in a Bernoulli trial to achieve the first success
is a random variable that follows the
*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

- Poisson distribution property: mean and variance are the same