suppose X has a binomial distribution with Pmf

Question

f(x,5,1/4)=(5 c x )(1/4)^x(3/4)^(5-x), x=0,1,2........:..,5

Write the Mgf and use it to find the mean and variance of the given disteibution.

Answer 1

The MGF is m[x](t)=(q+pe^t)n where q=1-p.

The first moment is derived by differentiating M wrt t:

M'(t)=n(q+pe^t)^(n-1).pe^t=npe^t(q+pe^t)^(n-1).

The first moment gives us the mean when t=0: M'(0)=np. With n=5 and p=1/4, mean=5/4=1.25.

We get the second moment by differentiating again:

Let u=npe^t and v=(q+pe^t)^(n-1); u'=npe^t and v'=(n-1)(q+pe^t)^(n-2).pe^t.

M"(t)=uv'+vu'=npe^t[(n-1)(q+pe^t)^(n-2).pe^t]+(q+pe^t)^(n-1)[npe^t]=
npe^t(q+pe^t)^(n-2)(q+npe^t).

When t=0, M"(0)=np(q+np).

Variance=M"(0)-(M'(0))^2=

np(q+np)-n^2p^2=npq=np(1-p)=(5/4)(3/4)=15/16.