Question

The mean and variance of binomial variate are 8 and 6. Find
the probability distribution function.​

Answer 1

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

Given that

• Mean of Binomial Distribution = 8

• Variance of Binomial Distribution = 6

Let assume that

• n be number of independent trials.

• p be probability of success

• q be probability of failure

We know,

In Binomial Distribution,

$\red{\rm :\longmapsto\:Mean = np}$

$\rm :\implies\:np = 8 - - - (1)$

And

$\purple{\rm :\longmapsto\:Variance = npq}$

$\rm :\implies\:npq = 6$

$\rm :\implies\:8q = 6 \: \: \: \: \: \: \: \: \{using \: (1) \: \}$

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

We know,

$\green{\bf :\longmapsto\:p + q = 1}$

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

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

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

$\bf :\longmapsto\:p = \dfrac{1}{4}$

On substituting the value of p, in equation (1), we get

$\rm :\longmapsto\:n \times \dfrac{1}{4} = 8$

$\bf\implies \:n = 32$

Thus,

We have now,

$\rm :\longmapsto\:n = 32$

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

$\rm :\longmapsto\:q = \dfrac{3}{4}$

Let 'r' be a random variable associated with the Binomial Distribution,

So,

Binomial Distribution is given by

$\bf :\longmapsto\:P(r) = \: ^nC_r \: {p}^{r} \: {q}^{n - r}$

On substituting the values of n, p and q, we get

$\rm :\longmapsto\:P(r) \: = \: ^nC_r \: { \bigg(\dfrac{1}{4} \bigg)}^{r} { \bigg(\dfrac{3}{4} \bigg)}^{32 - r} \: \: where \: 0 \leqslant r \leqslant 32$