Wave Functions Normalization

Often the relative probability of finding an atom in its excited state at time t is given by |psi(t)|^2 ~ e^(-2t/T), where T is the lifetime of the excited state. Normalize this probability distribution, and when does the probability drop to half the maximum value?

I integrated |psi(t)|^2 from 0 to T, and found that |psi(t)|^2 = e^(t-t/e^2). Is this right?

Also, I said therefore, the probability never drops to half the max. value, this doesn't seem correct to me..

Can someone check these out?

Thanks