Geometric Distribution
A discrete random variable is said to have a geometric distribution if
•There are independent and identical trials. Each trial can be thought of as a draw from a
population, where the drawing is done with replacement.
•Each trial has two possible outcomes, success and failure.
•The probability of success on each trial is the same, p. Therefore, the probability of failure
on each trial is 1 −p=q.
•The experiment is repeated until the first success occurs.
•The random variable Y is defined as the number of the trial on which the first success occurs.
•The parameter for the geometric random variable Y is the probability of success on each trial
p.
•The probability distribution function of the geometric random variable Y is
p(y) = p(1 −p)y−1for y= 1,2, . . .
•The theoretical mean of the geometric random variable Y is
µ=E(Y) = 1
p
•The variance of the geometric random variable Y is
σ2=V(Y) = 1−p
p2
Working with geometric random variables in R.
The R built-in function for the geometric distribution defines the random variable Ydifferently
than your textbook. To avoid any confusion, I have written functions in R called geo which use the
R built-in functions for the geometric distribution to calculate probabilities and generate random
values from the geometric random variable Yas defined in your textbook. Before you use R for the
geometric distribution, you will need to copy and paste the following functions into the command
window.
dgeo<- function(y,p){dgeom(y-1,p)}
rgeo<- function(n,p){rgeom(n,p) + 1}
To find a probability P(Y=y) = p(y) for a single value y, the command in R is
dgeo(y,p)
To find the probability P(Y≤y), use the sum command to add up all p(y) values for ybetween
and including 1 and y.
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