Question

Find the mean, variance, and standard deviation of the binomial distribution with the given values of n and p. n=50 p=0.2.

Answer 1

Concepts and reason

The concept of Binomial Distribution is used to calculate the mean, variance, and standard deviation.

Binomial distribution can be used when the outcomes of certain variables can be classified into two categories: success and failure. The outcomes should be independent of each other.

The mean of binomial distribution can be calculated by multiplying the number of trials and the probability of the success of each trial.

The variance can be calculated by multiplying the number of trials, the probability of success, and the probability of failure.

The standard deviation is the square root of the variance.

Fundamentals

There are n repeated trails in a binomial distribution. If there are n trials, the probability of getting x success among those n trails is defined as,

$P\left( {X = x} \right) = \left( \begin{array}{l}\\n\\\\x\\\end{array} \right){p^x}{\left( {1 - p} \right)^{n - x}}$

$\left( \begin{array}{l}\\n\\\\x\\\end{array} \right)$ is the number of combinations of getting $x$ successes in $n$ trials. The probability of success of each trail is denoted by $p$ . The formula to calculate the mean, variance, and standard deviation of binomial distribution are:

$\begin{array}{c}\\{\rm{Mean}} = np\\\\{\rm{Variance}} = np\left( {1 - p} \right)\\\\{\rm{Standard Deviation}} = \sqrt {Variance} \\\end{array}$

Here, $n$ is the number of trials and $p$ is the probability of the success of each trail. The probability of failure in each trail is denoted by $q = 1 - p$ .

The calculation of mean is:

$\begin{array}{c}\\{\rm{Mean}} = np\\\\ = 50 \times 0.2\\\\ = 10\\\end{array}$

The calculation of the variance is:

$\begin{array}{c}\\{\rm{Variance}} = np\left( {1 - p} \right)\\\\ = 50 \times 0.2 \times \left( {1 - 0.2} \right)\\\\ = 50 \times 0.2 \times 0.8\\\\ = 8\\\end{array}$

The calculation of the standard deviation is:

$\begin{array}{c}\\{\rm{Standard Deviation}} = \sqrt {Variance} \\\\ = \sqrt 8 \\\\ \approx 2.8284\\\end{array}$

Ans:

The mean of the binomial distribution with $n = 50$ and $p = 0.2$ is 10.