Expected ValueExpected Value
Expected ValueExpected Value
Math 115A
Spring 2008
Dawson
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Understanding Random Variables and Expected Value in Probability Theory - Prof. L. Dawson, Study notes of Mathematics
An introduction to the concept of random variables and expected value in probability theory. A random variable is a numerical value assigned to all possible outcomes of an experiment, and expected value is the long-term average value of a random variable when the experiment is repeated many times. Examples and calculations to help illustrate these concepts.
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Expected ValueExpected ValueExpected ValueExpected Value
Math 115ASpring 2008
Dawson
DescriptionDescription^
A random variable assigns a numericalvalue to all possible outcomes of anexperiment Often in probability we do not consider Often in probability we do not consider^ the actual outcomes, but rather weassociate numbers to events that arisefrom the experiment
ExampleExample^
Continuing with the above example,random variables give us an easier way todescribe events^ ◦
X
=3 is the event {TTT} ◦^
X
=3 is the event {TTT} ◦^
X
=2 is the event {THT, TTH, HTT} ◦^
X
= 1 is the event {THH, HHT, HTH} ◦^
X
=0 is the event {HHH} ◦^
X
≤ 2 is the event {THH, HTH, HHT, TTH,
THT, HTT}
ExampleExample^
We can also consider the probability of arandom variable, as we did before^ ◦
P
(X
◦^
P(
X
◦^
P(
X
◦^
P(
X
◦^
P(1 ≤
X
DescriptionDescription^
We can also look at
P
(X
x
P
(X
x
P
(X
x
P
(a
X
b
), etc.
◦^
P(
X
P
(X
◦^
P(
X
P
(X
DescriptionDescription^
It is very important whether we write anupper case (capital)
X
or a lower case
x
The capital letter represents a
random
variable
, and the lower case letter
variable
, and the lower case letter
represents an ordinary
number
ExampleExample^
Suppose you roll a fair die. If you roll a 1or 2 you lose $5, if you roll a 3 or 4 youwin $20, if you roll a 5 or 6 you don’t win^ or lose anything.or lose anything.^ ◦
Let
X
be the amount won on rolling the die
once.
So the possible values of
X
are -5, 20,
and 0.
ExampleExample
x^
P(
X
x)
x*P
(X
=
x)
2/6=1/
-5/
20
2/6=1/
20/
0
2/6=1/
0
E(X) = -5/3+20/3+
= 15/3 ≈ 5
DescriptionDescription^
The expected value of a random variable^ X
is often called the mean of
X
, denoted
mX
∑
=
⋅
=
=
n
i
i
X^
x X P x
X E^
)
(
) (
μ^
∑=
=
⋅
=
=^
i
i
i
X^
x X P x
X E^
1
)
(
) (
SummarySummary^
The expected value, E(X), of a randomvariable is the long-term average value ofX, when the experiment is repeated many^ timestimes It is often useful in business decisionswhen you want to weigh the risks oflosing money versus the chance of gains.
ExampleExample^ Let
X
represent the profit made.
Project I
x^
P(
X=
x)
x*
P(
X=
x)
-20,
.
-4,
0
.
0
Project II
200,
.
20,
x^
P(
X=
x)
x*
P(
X=
x)
-30,
.
-12,
0
.
0
300,
.
60,
ExampleExample^
Project I:^ ◦
E
(X
^
Project II
^
Project II^ ◦^
E(
X
ObjectivesObjectives^
Give the definitions of the terms
random
variable
and
parameter
^
Describe events using random variables. ^
Identify the possible values of a finite randomvariable defined on a sample space containing equally likely outcomes, and compute theequally likely outcomes, and compute the corresponding probabilities. ^
Explain the meaning of
E
(X
) and
m
X^
^
Compute the expected value of a finite randomvariable. ^
Find the value of a finite random variable (or aprobability) that would produce a given expectedvalue.