Question

If the mean.and
variance of a binomial
distribution with
parameters (n. p) are 40
and 30 respectively,

then parameters are
and
(40, 0.75)
(30, 0.25)
( (120, 0.5)
(160, 0.25)

Answer 1

$\large\underline{ \sf{Given- }}$

In a Binomial Distribution,

• mean = 40

and

• variance = 30

$\large\underline{\sf{To\:Find - }}$

• The parameters, n and p.

Concept Used :-

We know,

In a binomial distribution,

• n = number of trials

• p = probability of success

• q = probability of failure

• mean = np

• variance = npq

• p + q = 1

$\large\underline{\sf{Solution-}}$

It is given that

$\rm :\longmapsto\:mean \: of \: binomial \: distribution \: = \: 40$

$\bf\implies \:np = 40 - - (1)$

Also,

$\rm :\longmapsto\:variance \: of \: binomial \: distribution \: = 30$

$\bf\implies \:npq = 30$

$\rm :\longmapsto\:40q = 30 \: \: \: \: \: \: \: \: ( \because \: np \: = \: 40)$

$\rm :\implies\: \bf \: q \: = \: \dfrac{3}{4}$

As we know,

$\rm :\longmapsto\:p + q = 1$

So,

$\rm :\longmapsto\: \: p \: + \: \dfrac{3}{4} = 1$

$\bf\implies \:p \: = \: \dfrac{1}{4} - - - (2)$

On substituting equation (2) in equation (1), we get

$\rm :\longmapsto\:n \: \times \: \dfrac{1}{4} = 40$

$\bf :\implies\:n \: = \: 160$

$\rm \: Hence, \: the \: binomial \: parameter \: is \: (n, p) \: = (160, \: 0.25)$

Additional Information :-

1. Mean of Binomial Distribution is always greater than Variance.

2. The experiment consists of n identical trials.

3. Each trial results in one of the two outcomes, called success and failure.

4. The probability of success, denoted p, remains the same from trial to trial.

5. The n trials are independent.