Geometric Distribution
Consider repeated trials of a Bernoulli experiment e with probability P of success and q=1-P of failure. Let x be number of of times e must be repeated until finally obtaining a success. The distribution of random variable x is given as follows :
| k |
1 |
2 |
3 |
4 |
5 |
| P(k) |
P |
qP |
\[q^2P\] |
\[q^3P\]
|
\[q^4P\] |
The experiment e will be repeated k times only in the case that there is a sequence of k-1 failures followed by a success
P(k)= \[q^{k-1}P\]
The distribution is characterized by a single parameter P
Points to remember
Let X be the Geometric random variable with distribution GEO(P).Then
1. Expectation E[X]=\[\frac{1}{P}\]
2.Variance is given by Var[X]=\[\frac{q}{P^2}\]
3.Cumulative distribution F(k)= -1-\[q^k\]
4. P(x>r)= \[q^r\]
Example Suppose the probability that team A wins each game in a tournament is 60 percent. A plays untill it loses.
1. Find the expected value
2. find the probability P that A plays at least 4 games
3.Find the probability P that A wins a tournament if the tournament has 64 games.
solution 1. here P=0.4 and q=0.6
(a) E[X]= 1/P=2.5
(b) the only way A plays at least 4 games is if A wins the first 3 games .
Thus P=P(X>3)= \[q^3\]=\[0.6^3\]=0.216 =21.6%
(c) here A must win all the 6 games
So P=\[0.6^6=0.0467=4.67%\]
