The Geometric Distribution
Main Concept
The geometric distribution is a discrete memoryless probability distribution which describes the number of failures before the first success, x. The term also commonly refers to a secondary probability distribution, which describes the number of trials with two possible outcomes, success or failure, up to and including until the first success, x. This is known as the shifted geometric distribution. In this application, we will look closer at the first distribution.
Given a sequence of Bernoulli trials where p is the probability of success on each trial, the probability function for the geometric distribution is:
PY=x = p 1−p x, for x = 0, 1, 2, 3...
where x is the number of trials or failures before the first success.
The cumulative probability function is given by:
1−1−px+1
Properties
If p is the probability of a successful event, then:
Description
PY = x
PMF
The probability mass function.
p 1−p x
CDF
The cumulative distribution function.
Mean E(X)
The expected value of a random variable.
1 − pp
Variance Var(X)
Represented by the symbol σ2, this tells how much variation or spread exists from the mean value.
1−pp2
Example
Suppose a biased coin comes up head with a probability of 0.2 when tossed. What is the probability that you do not flip heads until the 1st, 2nd, 3rd,and 4th tosses?
Let the probability of success, p = 0.2.
We are looking for the probability that you do not flip heads until the first toss, which means that we are looking for the probability that there are exactly 0 failures before the first success:
P(0 failure before 1st success) = f(0) = P(X = 0) = 0.2 1−0.20 = 0.20
P(1 failure before 1st success) = f(1) = P(X = 1) = 0.2 1−0.21 = 0.16
P(2 failures before 1st success) = f(2) = P(X = 2) = 0.2 1−0.22 = 0.128
P(3 failures before 1st success) = f(3) = P(X = 3) = 0.2 1−0.23 = 0.1024
Change x, the number of trials, and p, the probability of success, to observe the change in the geometric distribution.
# of trials before success (x) =
probability of success (p) =
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