Question

Let \( X \) and \( Y \) be two positive integer-valued random variables with the joint probability mass function \[ P(X = m, Y = n) = \begin{cases} g(m) h(n), & m, n \geq 1, \\ 0, & \text{otherwise}. \end{cases}\] where \( g(m) = \left( \frac{1}{2} \right)^{m-1} \), \( m \geq 1 \), and \( h(n) = \left( \frac{1}{3} \right)^n \), \( n \geq 1 \). Then \( E(XY) \) equals .......... 
 

Hint:

For expected values of products of independent random variables, separate the sums and use known results for geometric series.

Answer 1

Correct Answer: 2.5 - 3.5

The joint pmf is
\[P(X=m,Y=n)=g(m)h(n),\qquad m,n\ge1,\]
with
\[g(m)=\left(\tfrac12\right)^{m-1},\qquad h(n)=\left(\tfrac13\right)^n.\]
Thus \(X\) and \(Y\) are independent, so
\[E(XY)=E(X)\,E(Y).\]

Compute \(E(X)\)

\[E(X)=\sum_{m=1}^\infty m\left(\tfrac12\right)^{m-1} =\frac{1}{(1-\tfrac12)^2}=4.\]

Compute \(E(Y)\)

\[E(Y)=\sum_{n=1}^\infty n\left(\tfrac13\right)^n =\frac{\tfrac13}{(1-\tfrac13)^2} =\frac34.\]

Final value

\[E(XY)=4\times\frac34=3.\]

\[\boxed{E(XY)=3}\]