Let X and Y be discrete random variables. Let E[X] and var[X] be the expected value and variance, respectively, of a random variable X. (a) Show that E[X + Y ] = E[X] + E[Y ]. (b) If X and Y are independent, show that var[X + Y ] = var[X] + var[Y ].

(a)

(b)

Step-by-step explanation:

Let X and Y be discrete random variables and E(X) and Var(X) are the Expected Values and Variance of X respectively.

(a)We want to show that E[X + Y ] = E[X] + E[Y ].

When we have two random variables instead of one, we consider their joint distribution function.

For a function f(X,Y) of discrete variables X and Y, we can define

Since f(X,Y)=X+Y

Let us look at the first of these sums.

Similarly,

Combining these two gives the formula:

Therefore:

(b)We  want to show that if X and Y are independent random variables, then:

By definition of Variance, we have that:

Since X and Y are independent, Cov(X,Y)=0

Therefore as required: