Calculating Expected Values
Dennis Sun
Expected Value Formula
We typically write the p.m.f. of a random variable as a formula:
n x
p[x] =
p (1 − p)n−x
x
p[x] = (1 − p)x−1 p
So to calculate the expected value of a random variable X with
p.m.f. p[x], we just have to evaluate the sum
X
E[X] =
x · p[x].
x
Expected Value of a Binomial Random Variable
Let’s find the expected value of a Binomial(n, p) random variable
X.
E[X] = np
Expected Value of Functions of R.V.s
How do you calculate the expected value of a function of a random
variable, g(X)?
There are two ways:
1
Find the PMF of Y = g(X) and calculate
X
E[Y ] =
y · pY [y].
y
2
Use the Law of the Unconscious Statistician (LotUS).
X
E[g(X)] =
g(x) · pX [x].
x
Example
You roll a die. If it lands
, you win $1.
# lose $1.
"If it lands , you
2
amount
.
Otherwise, nothing happens. Find E
you win
Notice that E
"
amount
you win
2 #
amount
6 E
=
you win
2
.
Many students make the mistake of thinking that E[g(X)] is equal
to g(E[X]).
This is only true when g is a linear function, i.e., g(X) = aX + b. In
this case, E[aX + b] = aE[X] + b.