The mean and variance of a random variable having a binomial distribution are 4 and 2 respectively, then is Option 1) 1/16 Option 2) 1/8 Option 3) 1/4 Option 4) 1/32

As we learnt in

Binomial Distribution(Statistical) -

-

Mean=np=4 ........ (1)

Variance=np(1-p)=2 .............. (2)

From (1) and (2),

From (1), n=8

Thus, P(X=1)=

Option 1)

1/16

This option is incorrect

Option 2)

1/8

This option is incorrect

Option 3)

1/4

This option is incorrect

Option 4)

1/32

This option is correct

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As it is give in the question that:

mean(np)=4 (equation.....1)

variance(npq)=2(equation....2)

now divided equ(2)/equ(1);then:

npq/np=2/4

q=1/2

now as we know that sum of the probability=1

p+q=1

p+1/2=1

p=1-1/2

p=1/2

now substitute p value in equation (1)

np=4

n(1/2)=4

n=4*2

n=8

now as we know that p(X=1)

by the general formula p(X=R)=ncr (p)^r (q)^n-r

now from th general formula above:

p(X=1)=8c1(1/2)^ (1/2)^8-1

=8*(1/2)^7+1

=8*(1/2)^8

=8*1/256

p(X=R) =1/32

therefore this is the whole procedure for the above question