Question

Question:

(a) What does it mean to say that a random variable has a geometric distribution with parameter \(p\) ?

b) What is the mean of a geometric distribution with parameter \(p\) ?

Short answer

Answer

(a) The probability distribution of the first success among independent trials with a constant probability \(p\) of success is the resultant answer.

(b) The resultant answer is\(\frac{1}{p}\).

Step-by-step solution

Given data

The given data is that a random variable has a geometric distribution.

Concept of Geometric probability

Geometric probability: \(P(X = k) = {q^{k - 1}}p = {(1 - p)^{k - 1}}p\).

Simplify the expression 

(a)

A random variable that has a geometric distribution with probability \(p\) of success represent the probability distribution of the first success among independent trials with a constant probability \(p\) of success.

Definition geometric probability: \(P(X = k) = {q^{k - 1}}p = {(1 - p)^{k - 1}}p\) that is, the probability of the first success occurring on the \(k - th\) trial is \({(1 - p)^{k - 1}}p\).

Find the geometric distribution

(b)

The expected value of a geometric distribution is the reciprocal of the probability of success \(p\): \(\mu = \frac{1}{p}\).