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Chapter 5 Discrete Probability
Distributions
5.3
5.4
EXPECTATION
5.3.1 The Mean and Expectation
(Expected Value)
5.3.2 Some Applications
VARIANCE AND STANDARD
DEVIATION
5.3 EXPECTATION
5.3.1

The Mean and Expectation
(Expected Value)
Experimental approach


Scroe, x
1
2
3
4
5
6
Frequency, f
15
22
23
19
23
18

6
fx
x
Suppose we throw an unbiased die 120 times
and record the results:
= i 1
i
i 1

=
6
f
i
Then we can calculate the mean score obtained
where
i
= ________ (3 d.p.)

Theoretical approach

The probability distribution for the random
variable X where X is ‘the number on the die’ is
as shown:
Score, x
1
2
3
4
5
6
P(X = x) 1/6

We can obtain a value for the ‘expected mean’
by multiplying each score by its corresponding
probability and summing, so that
Expected mean =
=

If we have a statistical experiment:


a practical approach results in a frequency
distribution and a mean value,
a theoretical approach results in a probability
distribution and an expected value.
xP( X  x)

The expectation of X (or
expected value), written E(X) is
all x
given by
E(X) =

Example 1

random variable X has a probability
function defined as shown. Find E(X).
x
-2
-1
0
1
2
P(X= x)
0.3
0.1
0.15
0.4
0.05

In general, if g(X) is any function of the
discrete random variable X then
In general, if g(X) is any function of the discrete
random variable X then
E[g(X)] =
 g ( x) P( X  x)
all x

Example

In a game a turn consists of a tetrahedral die
being thrown three times. The faces on the die
are marked 1,2,3,4 and the number on which the
2
x
die falls is noted. A man wins $
whenever x
fours occur in a turn. Find his average win per
turn.

Example

x
The random variable X has probability function P(X
= x) for x = 1,2,3.
x
1
2
3
P(X = x)
0.1
0.6
0.3


Calculate (a) E(3), (b) E(X), (c) E(5X), (d) E(5X+3),
(e) 5E(X) + 3, (f) E(X2), (g) E(4X2- 3), (h) 4E(X2 ) – 3.
Comment on your answers to parts (d) and (e) and parts
(g) and (h).
E(a X + b) = a E(X) + b, where a and b are any constants.
E[f1(X)  f2(X)] = E[f1(X)]  E[f2(X)], where f1 and f2 are functions of X.

5.3.2
Some Applications
5.4 VARIANCE AND
STANDARD DEVIATION
The variance of X, written Var(X), is given by
Var(X) = E(X - )2