Let W1W2 be iid with distribution Exponential 3 Prove that f

Let W1,W2, be i.i.d with distribution Exponential (3). Prove that for some n, we have P(W1+W2+...+Wn<n/2) >.999

Solution

Since, W1,W2, be i.i.d with distribution Exponential (3), we know that W1+W2+...+Wn is Gamma(n,1/3). Therefore, from the tables of gamma distribution, we can see that for all n>48, the quired probability is P(W1+W2+...+Wn<n/2) >.999