Geometric Distribution

A geometric distribution is defined as a discrete probability distribution of a random variable “k” which determines some of the conditions.

1. An event that has a series of trails.

2. Each trial has only two possible results i.e. either success or failure.

3. The probability of success is similar for each trail.

In geometric and statistics, geometric distribution states the probability that first success appears after y number of trials. If p is the probability of success or failure of each trial, then the probability that success appears on the yth trial is derived by the formula.

Pr( X = y) = \[(1-p)^{y-1}\]p

Geometric Distribution Examples

Some of the geometric distribution real-life examples are given below:

1. A person is looking for a job that is both challenging and satisfying. What is the probability that he will drop out zero times, one time, two times, and so on unless he gets his new job?

2. Examine a couple who are planning to conceive a child and they will proceed with babies unless it is a girl. What is the probability that the couple has zero boys, one boy, two boys, and so on unless a girl is delivered?

3. A pharmaceutical company is planning to design a new drug to treat a certain disease that will have minimum side effects. What is the probability that zero drugs failed the test, one drug failed the test, two drugs failed the test, and so on unless they come up with the newly designed ideal drug.?

Mean of Geometric Distribution

Each probability distribution has its particular formula for mean and variance of the random variable x. The mean of the expected value of x determines the weighted average of all possible values for x. For a mean of geometric distribution E(X) or μ is derived by the following formula.

E(Y) = μ = 1/P

Solved Examples

1. Find the probability density of geometric distribution if the value of p is 0.42; x = 1,2,3 and also calculate the mean and variance.

Solution:

Given that p = 0.42 and the value of x = 1, 2, 3

The formula of probability density of geometric distribution is

P(x) = p (1-p) x-1; x =1, 2, 3

P(x) = 0; otherwise

P(x) = 0.42 (1- 0.42)

P(x) = 0; Otherwise

Mean= 1/p = 1/0.42 = 2.380

Variance = 1-p/ p2

Variance = 1-0.42 /0.422

Variance = 3.287

2. If the probability of breaking the pot in the pool is 0.4, find the number of brakes before success and the corresponding variance and standard deviation.

Solution: 

Here,

X ∼ geo(0.4)

Hence,

e(x) = 1/0.4 = 2.5

We can expect to pot off the break after 2.5 goes.

Var(x) = 0.6/0.4²

= 3.75

Hence, standard deviation ( σ) = 1.94

Quiz Time

1. Which of the following statements is true about geometric distribution?

1. Geometric distribution is approximately skewed right

2. Geometric distribution is  approximately symmetric

3. Geometric distribution is  approximately skewed left

4. The shape can be either symmetric or skewed.

2. Which of the following statements is not true about the geometric distribution?

1. Trails should be fixed

2. The probability of success should be 0.5

3. Events should be independent

4. Their distributions are approximately symmetric.

 3. What will be the variance of geometric distribution having parameter p = 0.72?

1. 54%

2. 76%

3. 13%

4. 69%