Question

What is the formula for the variance of a geometric distribution?

Answer 1

Hint: To solve this question we need to have the concept of geometric distribution and binomial probability. The normal probability is basically this statistical function having just two outcomes. These outcomes are either success or failure. The Variance is (σ2), is defined as the sum of the squared distances of each term in the distribution from the mean (μ), divided by the number of terms in the distribution (N).

Complete step by step solution:
The question asks us to find the formula for the variance of geometric distribution. A geometric probability distribution describes one of the two discrete probability situations. The probability distribution is a statistical function that describes all the possible values and that a random variable can take within a given range.
Geometric distribution is related to binomial probability. It basically represents the number of failures before we get a success in a series of Bernoulli trials. To get the formula let us consider a case of binomial trial. Since we know that the binomial trial has two outcomes only, which is a success or failure. So let us assume ‘p’ as the probability of success and ‘q’ as the probability of failure. So when written in case of binomial probability $p$ is the difference of $1$ and $q$, which in mathematical form is shown as:
$\Rightarrow p=1-q$
Now, supposedly the trial is conducted ‘X’ number of times and the first success occurs on $k^{th}$ time. If the probability of such occurrence can be expressed as some geometric function of ${}^{″}{p}^{″}$ then the probability distribution is called geometric distribution probability which in short is written as $\text{''gdf ''}$ . If the above fact is written in mathematical form it becomes $\text{P}\left(\text{X=k}\right)\text{ = gdf}\left(\text{p}\right)$ .
$\therefore$ The formula for the variance of a geometric distribution ${\sigma }^2={\displaystyle \frac{1-p}{p^2}}$ .

Note: Mean in geometric distribution is the ratio of $1-p$ to $p$ and standard deviation as the square root of the ratio of the mean and number of successes. Mathematically it is represented as:
Mean = ${\displaystyle \frac{1-p}{p}}$ and Standard Deviation = ${\sigma }^2=\sqrt{{\displaystyle \frac{1-p}{p^2}}}$