An individual taken from a very large biological population is of type A with probability p=0.5(1+t) and of type B with probabiliy 1-p=0.5(1-t).

a) Suppose that X denotes the number of type A individuals in a random sample of size n. What is the probability that X = x?

My answer: $\displaystyle 0.5^x*(1+t)^x*(1-t)^{(n-x)}$ Classic binomial situation.

b)Find the maximum likelihood estimator of t and show that it is unbiased.

My answer: Find log likelihood, differentiate yields

2x-n(1+t)/[(1+t)(1-t)]

setting to zero yields t=(2x-n)/n

To show unbiased, I need to find E(t). This is an integral including $\displaystyle 0.5^x*(1+t)^x*(1-t)^{(n-x)}$ which I have no idea how to integrate.

Thanks for your help.