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Expected Value of a Random Variable
Dennis Sun
The “Center” of a Random Variable
p[x]
Consider the number of tails in 5 tosses of a fair coin. Its
distribution looks like
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00 0
1
2
x
3
4
5
This distribution seems to be centered at 2.5. Can we say that the
“average” or “mean” number of tails is 2.5?
Expected Value
Yes! The “mean” of a random variable X is called its expected
value. We write it as E[X].
Caution! The expected value may not actually be a possible value.
For example, the expected number of tails in 5 tosses is 2.5, even
though it’s not actually possible to get 2.5 tails.
Weighted Average
• Suppose I ask you to calculate the average of the numbers: 1,
1, 2, 4, 4, 4, 4, 5, 5, 6.
1+1+2+4+4+4+4+5+5+6
• Approach 1:
= 3.6.
10
• Approach 2: Sum over the distinct numbers, weighting by
how often each number occurs:
1·
2
1
4
2
1
+2·
+4·
+5·
+6·
= 3.6.
10
10
10
10
10
• We can write the two formulas as
Pn
y=
i=1 yi
n
=
X
x
x · wx .
Expected Value
The expected value is a weighted average. We weight each
possible value by its probability of occurring:
X
E[X] =
x · p[x].
x
Interpretation: The expected value is the long-run average if we
were to observe many realizations of a random variable.
Example
p[x]
Let’s check that E
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00 0
h
number
of tails
i
= 2.5.
x
0
1
2
3
4
5
1
2
x
3
4
5
p[x]
.03125
.15625
.3125
.3125
.15625
.03125