Some Random variables proofs.

1) I've 2 random variables X ~ Bin(n,p), Y ~Bin(n, 1-p) Now I need to show that Fx(i) = 1 - Fy(n-i-1) where F is the normal binome distribution. Now I've tried just putting i and I get that Fy(n-i-1) = Fx(i+1) and beats me how Fx(i) + Fx(i+1) = 1.

2) I need to find a $\lambda$ so that Fx(k) is maximal for X ~ Poi($\lambda$) meaning the poissan distribution.

Thank you!

• Note that binomial distributions are discrete, and that $F(i)$ is a probability of ${0,1,2,3,...,i}$ with $i$ included. Commented Dec 6, 2016 at 11:32
• Thanks for the reply although I don't see how that helps me, since F(i) is the probabilty of 0,1,2,3,..i and F(i+1) is 0,1,2,3,4,...i+1 how exactly do their sum is 1? Commented Dec 6, 2016 at 11:35
• $1 - Fy(n-i-1) = Fx(i)$, because of the weak inequality. $Fx(i) = Bin(n,p) [0,i]= Bin(n,1-p) [n,n-i]$ Commented Dec 6, 2016 at 12:37
• Okay thanks! got it. Commented Dec 6, 2016 at 13:17

Concerning your first question: It is simpler than you think. Just move the CDF expression of the random variable $Y$ to the other side of the equation. Once they are on the same side write both functions explicitely as sums. Open the two sigma expressions and observe the addend components within. Does this overall sum look familiar to you?