We have seen that X 1 + X 2 is a Bin(n 1 + n 2 , p) distributed random variable. Viewing X 1 + X 2 + X 3 as the sum of X 1 + X 2 and X 3 , it follows that X 1 + X 2 + X 3 is a Bin(n 1 + n 2 + n 3 , p)...

We have seen that X₁
+ X₂
is a Bin(n₁
+ n₂, p) distributed random variable. Viewing X₁
+ X₂
+ X₃
as the sum of X₁
+ X₂
and X₃, it follows that X₁
+ X₂
+ X₃
is a Bin(n₁
+ n₂
+ n₃, p) distributed random variable.

2. The sum rule for two normal random variables tells us that X + Y is a normally distributed random variable. Its parameters are expectation and variance of X + Y . Hence by linearity of expectations

µX+Y = E[X + Y ] = E[X]+E[Y ] = µX + µY =3+5=8

and by the rule for the variance of the sum

using that Cov(X, Y ) = 0 due to independence of X and Y