 # 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%

1. What are the Applications of Geometric Distributions?

The geometric distribution is used to examine the possibility of success given a limited number of trials which are highly used in real-world environments where the unlimited trials are rare. Some of the applications of geometric distribution in reals-life are as follows:

1. The geometric distribution in sports primarily, in baseball, is used to examine the probability of a batter earning a hit before he gets three strikes, here the aim is to reach towards success within three trials.

2. Geometric distribution in time management  is used to reach a success before a specific period of time.

3. The geometric distribution is primarily used by the  companies in cost-benefit analysis. The company decides whether the fund reaches trials, if successful, will the company be able to earn estimated profit, the aim is to reach success before the cost outweighs the potential gain.

The geometric distribution is applied on an instinctive level in daily life regularly.

2.  What are the Properties of Geometric Distribution in Statistics?

There are multiple values that provide information about a specific probability distribution.

1. The mean or expected value of distribution provides us information about what average one would expect from multiple numbers of trials

2. The mode of distribution is the value that has the highest  possibility of appearing.

3. The median of a distribution is another measure of central tendency, useful when the distribution includes an outlier( i.e. specifically large values /small values) that makes the mean confusing.

4. The variance of distribution determines the way of “ spread out” data.

The three values mean, mode and variance are generally used for geometric distribution whereas the median is not generally determined.