# Suppose that X_{1}, X_{2} and X_{3} are three independent random variables with the same distribution as X. What is the ecpected value of the sum X_{1

Random variables
Suppose that $$X_{1}, X_{2} and X_{3}$$ are three independent random variables with the same distribution as X.
What is the ecpected value of the sum $$X_{1}+X_{2}+X_{3}$$? The product $$X_{1}X_{2}X_{3}$$?
Suppose a discrete random variable X assumes the value $$\frac{3}{2}$$ with probability 0.5 and assumes the value $$\frac{1}{2}$$ with probability 0.5.

2021-05-13
Step 1
The expectation of X is as follows:
$$E(X)=\mu$$
$$=\sum xP(X=x)$$
$$=\frac{3}{2}\times (0.5)+\frac{1}{2}\times (0.5)$$
=1
Step 2
The random variables $$X_{1}, X_{2} ,and X_{3}$$ are independent and have the same the distribution of X. The required expectations are calculated as follows:
$$E(X_{1}+X_{2}+X_{3})=E(X_{1})+E(X_{2})+E(X_{3})$$
=E(X)+E(X)+E(X)
=3E(X)
$$=3\times 1$$
=3
$$E(X_{1}X_{2}X_{3})=E(X_{1})\times E(X_{2})\times E(X_{3})$$
$$=E(X)\times E(X)\times E(X)$$
$$=1\times 1\times 1$$
=1