Question

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
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asked 2021-05-12
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.

Answers (1)

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
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