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

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