From Algorithms to Z-Scores - matloff - University of California, Davis

3.4. EXPECTED VALUE 41
• For random variables X and Y—not necessarily independent—and constants a and b, we have
E(aX + bY ) = aEX + bEY (3.15)
This follows by taking U = aX and V = bY in (3.13), and then using (3.15).
• For any constant b, we have
E(b) = b (3.16)
For instance, say U is temperature in Celsius. Then the temperature in Fahrenheit is W = 9
5U +32.
So, W is a new random variable, and we can get its expected value from that of U by using (3.15)
and b = 32.
with a = 9
5
Another important point:
Property D: If U and V are independent, then
E(UV ) = EU · EV (3.17)
In the dice example, for instance, let D denote the product of the numbers of blue dots and yellow
dots, i.e. D = XY. Then
E(D) = 3.5 2 = 12.25 (3.18)
Equation (3.17) doesn’t have an easy “notebook proof.” It is proved in Section 8.3.1.
Consider a function g() of one variable, and let W = g(X). W is then a random variable too. Say
X takes on values in A, as in (3.7). Then W takes on values in B = {g(c) : cɛA}. Define
Then
so
Ad = {c : c ∈ A, g(c) = d} (3.19)
P (W = d) = P (X ∈ Ad) (3.20)