Question

Given $X - B(n,p)$
$$If $n = 10$and$p = 0.4$,find $E(X)$and$Var(X)$

Answer 1

Hint:-In this question first we need to find the probability of failure outcome using the relation that the sum of successful and failure outcome equal to one. Then , we have to put successful and failure outcomes in formulas of mean and variance to get the answer.

Complete step-by-step answer:

Binomial Distribution:- It is a specific probability distribution. It is used to model the probability of obtaining one of two outcomes, a certain number of times (k), out of a fixed number of trials(N) of a discrete random event. A binomial distribution has only two types of outcomes:
Success outcome: The expected outcome is called success. The probability of a successful outcome is denoted by “p”.
Failure outcome: Any other outcome other than success outcome is called failure outcome.
It is denoted by “q” or “1-p”.

The mean of the binomial distribution is np.
$ \Rightarrow E(X) = np$ eq.1
 and the variance of the binomial distribution is np (1 − p).
$ \Rightarrow Var(X) = npq$ eq.2
Since, we have probability of success$p = 0.4$.Then probability of failure is
$
   \Rightarrow p + q = 1 \\
   \Rightarrow q = 1 - p \\
   \Rightarrow q = 1 - 0.4 \\
   \Rightarrow q = 0.6{\text{ eq}}{\text{.3}} \\
$
Now , mean of the given binomial distribution$E(X)$
$
   \Rightarrow E(X) = np \\
   \Rightarrow E(X) = 10 \times 0.4 \\
   \Rightarrow E(X) = 4 \\
$
Therefore, the mean of the given binomial distribution is 4.

Now, the variance of the given binomial distribution$Var(X)$
$
   \Rightarrow Var(X) = npq \\
   \Rightarrow Var(X) = 10 \times 0.4 \times 0.6 \\
   \Rightarrow Var(X) = 2.4 \\
$
Hence, the variance of the given binomial distribution is 2.4.
Note:-Whenever you get this type of question the key concept to solve this is to evaluate all the variables involved in the formula which is used in the given question like in this question we need to find $q$ only, And simply use the formulas of mean and variance of the binomial distribution.