1MA01: Probability
Sinéad Ryan
TCD
November 29, 2013
Expected value
The expected value of a discrete random variable X is
X
E(X) =
kP(X = k)
k
where the sum is over all possible values of k.
Also called the mean of X and sometimes written
μ = E(X).
Example 1: Roll a die. Let X be the number rolled. Find
μ.
by P.E.L.O. P(X = 1) = P(X = 2) = . . . = P(X = 6) = 61 . Then
μ =
6
X
kP(X = k)
k=1
= 1
‚ Œ
1
= 3.5
6
+2
‚ Œ
1
6
+3
‚ Œ
1
6
+4
‚ Œ
1
6
+5
‚ Œ
1
6
+6
‚ Œ
1
6
(and since each number is equally likely this is just the
average.) a histogram looks like:
P(X)
1/6
1
2
3
4
X
5
6
Example 2: Roll a die once. If it lands on 6 you win $4,
otherwise you lose $1. . Find μ.
Let M be the amount won.
μ = E(M) = 4(1/ 6) + (−1)(5/ 6) = −1/ 6 ∼ −$0.167.
A histogram looks like:
0.8
P(X)
0.6
0.4
0.2
0
-1
4
µ=-0.167
M
Variance and Standard Deviation
Definition
Let X be a discrete random variable with expected value,
μ = E(X) then
the variance is
Var(X) = E[(X − μ)2 ] =
X
(k − μ)2 P(X = k)
k
where the sum is taken over all possible values of X.
the standard deviation of X is
p
SD(X) = Var(X)
the symbol σ is commonly used for SD(X).
Note that
the variance is a sum of terms that are products of
two positive numbers ie (k − μ)2 is always positive
because of the square and P(X = k) is positive since
it is a probability.
the term (k − μ)2 can be thought of as the square of
the distance between the average value μ and k.
an example
Roll a die. Let X be the number rolled. Find Var(X) and
SD(X)
Recall for the same experiment we showed earlier that
μ = 72 . Then
Var(X)
=
6
X
(k − μ)2 P(X = k)
k =1
‚
Œ2 ‚ Œ
1
‚
7
Œ2 ‚ Œ
1
‚
7
Œ2 ‚ Œ
1
+ 2−
+ 3−
6
2
6
2
6
‚
Œ2 ‚ Œ ‚
Œ2 ‚ Œ ‚
Œ2 ‚ Œ
7
1
7
1
7
1
= + 4−
+ 5−
+ 6−
2
6
2
6
2
6
35
=
.
12
=
1−
7
2
The standard deviation is σ = SD(X) =
Æ
35
12
≈ 1.708.
The calculation of Var can be simplifed.
Theorem
Let X be a discrete random variable with expected value
μ = E(X). Then
X
k 2 P(X = k)
Var(X) = E(X2 ) − μ2 with E(X2 ) =
k
As an exercise check that in the previous example
!
6
X
35
2
Var(X) =
k P(X = k) − μ2 =
12
k=1
Expected value of binomial distributions
For binomial (n,p) distributions the definitions for Var(X)
and SD(X) given above work but can be simplified
E(X) = np
Var(X) = npq
p
SD(X) =
npq