# Geometric Distribution

#### galactus

##### New Member
A geometric Distribution is a discrete probability distribution of a random variable x that satisifies:

1. A trial is repeated unitl a success occurs

2. The trials are independent.

3. The probability is constant for each trial.

Many things are repeated until a success occurs. You may take your driver's exam several times before you pass and acquire your driver's license. You may attempt to dial your internet connection many times before successfully logging on. You can use a geometric distribution in such scenarios.

The probability that the first success will occur on trial x is:

P(x)=p(q)^(x-1), where q=1-p

Let's look at an example:

"A music store finds that 1 in every 100 CD players it buys from a supplier is defective. Find the probability that the first defective unit is the 10th sold.

1/100=0.01=p

q=1-p=0.99

x=4

(0.01)(0.99)^9=0.009

"Find the probability that the 1st defect is the 1st, 2nd, or 3rd player sold":

We add up P(1), P(2), P(3).

(0.01)(0.99)^(0)+(0.01)(0.99)^(1)+(0.01)(0.99)^(2)=0.29701

Even though, in theory, a success may never occur, the geometric distribution is discrete because x can be listed as 1, 2, 3,.4, 5,........

As x becomes larger, P(x) gets closer and closer to 0.

The sum of all the probabilities is a geometric series with an infinite number of terms. The sum of a geometric series is given by S=a/(1-r), where a is
the first term in the series, r is the common ratio.

In this case, a=p, r=q and S=p/(1-q)=p/p=1