How to determine the expected value of Z when given a joint frequency function for X and Y, if Z = 2x +10?

Question

Suppose we have a frequency function for two variables X and Y, $f(x,y) = \frac{(x+y)}{30}$, for $x = 0, 1, 2$ and $y = 0, 1, 2, 3$.

I was initially asked to determine the values of $E(x)$ and $E(Y)$, which I did using a marginal distribution table:

$$E(x) = 0(1/5) + 1(1/3) +2(7/15) = 19/15$$ $$E(y) = 0(1/10) + 1(1/5) + 2(3/10) + 3(2/5) = 2$$

I was then asked to determine $E(X + Y)$.

$$E(X + Y) = E(X) + E(Y)$$ $$E(X + Y) = 19/15 + 2 = 49/15$$

Finally, if $Z = 2X + 10$, determine $E(Z)$. I know the values of Z would be 10, 12 and 14, but have no idea where to go from there.

Answer 1

Observe that for any two random variables $X$ and $Y$ and for any real number $a$ we have $$\Bbb E(aX+Y) = a\ \Bbb E(X) + \Bbb E(Y).$$

This property of the expectation is called linearity.

Also note that for any degenerate random variable $X=a$ (say) we have $$\Bbb E(X) = a.$$

Now use these linearity and degeneracy properties of the expectation to find $\Bbb E(Z)$ as follows $:$

$$\begin{align} \Bbb E(Z) & = \Bbb E(2X+10). \\ & = 2\ \Bbb E(X) + 10. \\ & = 2 \times \frac {19} {15} + 10. \\ & = \frac {188} {15}. \end{align}$$

Answer 2

Note that $$ EZ=2EX+10 $$ by linearity of expectation.