4.5Properties of the Expected Value 115
That is, the expected value of a constant is just its value. (Is this intuitive?) Also, if we take
b=0, then we obtain
E[aX]=aE[X]
or, in words, the expected value of a constant multiplied by a random variable is just the
constant times the expected value of the random variable. The expected value of a random
variableX,E[X], is also referred to as themeanor thefirst momentofX. The quantity
E[Xn],n≥1, is called thenth moment ofX. By Proposition 4.5.1, we note that
E[Xn]=
∑
x
xnp(x)ifXis discrete
∫∞
−∞
xnf(x)dx ifXis continuous
4.5.1 Expected Value of Sums of Random Variables
The two-dimensional version of Proposition 4.5.1 states that ifX andY are random
variables andgis a function of two variables, then
E[g(X,Y)]=
∑
y
∑
x
g(x,y)p(x,y) in the discrete case
=
∫∞
−∞
∫∞
−∞
g(x,y)f(x,y)dx dy in the continuous case
For example, ifg(X,Y)=X+Y, then, in the continuous case,
E[X+Y]=
∫∞
−∞
∫∞
−∞
(x+y)f(x,y)dx dy
=
∫∞
−∞
∫∞
−∞
xf(x,y)dx dy+
∫∞
−∞
∫∞
−∞
yf(x,y)dx dy
=E[X]+E[Y]
A similar result can be shown in the discrete case and indeed, for any random variablesX
andY,
E[X+Y]=E[X]+E[Y] (4.5.1)
By repeatedly applying Equation 4.5.1 we can show that the expected value of the sum
of any number of random variables equals the sum of their individual expectations.