Last modified: July 15, 2024
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Geometric Distribution (Discrete)
A discrete random variable X follows a geometric distribution if it represents the number of trials needed to get the first success in a sequence of Bernoulli trials. The geometric distribution is denoted as X∼Geometric(p), where p is the probability of success on each trial.
Probability Mass Function (PMF)
The PMF of a geometric distribution is given by:
P(X=k)=(1−p)k−1p
where k∈1,2,3,…, representing the number of trials until the first success.
Cumulative Distribution Function (CDF)
The CDF of a geometric distribution is given by:
F(k)=P(X≤k)=1−(1−p)k
Expected Value and Variance
The expected value (mean) of a geometric distribution is:
E[X]=1p
The variance of a geometric distribution is:
Var(X)=1−pp2
Moment Generating Function (MGF)
The moment generating function (MGF) of a geometric distribution is:
MX(t)=pet1−(1−p)et
Example: Flipping a Coin
Consider flipping a fair coin (probability of heads p = 0.5) until the first head appears.
I. What is the probability that the first head appears on the 3rd flip?
For the first head on the 3rd flip:
P(X=3)=(1−0.5)3−1×0.5=0.125
II. What is the expected number of flips to get the first head?
Expected number of flips:
E[X]=10.5=2
III. What is the variance in the number of flips?
Variance in the number of flips:
Var(X)=1−0.50.52=2
Applications
Geometric distributions are useful in scenarios where you are waiting for the first success in a series of independent trials, such as in quality control testing, network packet delivery analysis, and reliability testing in engineering.