Geometric Distribution

Definition

Given random variable :

where is the number of trials required to observe the first success in a sequence of independent Bernoulli trials with a probability of success .

The geometric distribution is a special case of the Negative Binomial Distribution, where .

As opposed to the Binomial Distribution, where we count the number of successes in trials, here we count the number of trials to achieve success.

An alternate formulation is where is the number of trails before first success (so one less than )

Support and PMF

(support and p.m.f.)

Support
1 S
2 FS
3 FFS
PMF

In a binomial experiment, the probability of a success on the th trial is:

where:

  • is the probability of success
  • is the probability of failure

Note that we have failures before success on the th trial.

Expectation and Variance

(expected value and variance)

Definition

Given :

Proof
1
2
3

See geometric series. Pretty cool.

Memorylessness

Property

The geometric distribution is memoryless

Example

Suppose we flip a fair coin.

Now suppose the first 6 tosses are tails (TTTTTT) which we will call event . Find .

solution

Examples

Example

What is the probability that an average Jeopardy player will appear in 75 shows?

solution
3 contestants, keep winning until first loss. .

We want him to win 74, then lose, so .