Solved: Suppose that Y is a binomial random variable based

Solution :

Step 1 of 5:

Let Y is a binomial random variable based on n trials with success probability p and

Our goal is:

a). We need to prove that E()= nq and V()= npq.

b). We need to show that show that .

c). We need to find based on the result in part (b), what is the distribution of ?

d). If Y is interpreted as the number of successes in a sample of size n. What is the    

     interpretation of  ?

e). Based on your in part D. Why are the answers to parts (a), (b), (c), “obvious”?  

a).

Now we have to prove that E()= nq and V()= npq.

We assume that .

Let .

Here a=-1 and b=n

For binomial distributed variable Y is

                                E(Y)=np and V(Y)=npq

Then E() is

E()=E()

E()= E(n-Y)

E()= n-E(Y)

We know that E(Y)=np.

E()= n-np

E()= n(1-p)

We know that q=1-p

E()= nq

Therefore E() is nq.

Now V() is

V()= (

Here .

V()= V(Y)

We know that V(Y)=pq.

V()= npq

Therefore V() is npq.