Download Steps for the Mean of a Probability Distribution

Unit 5
Section 5-3
5-3: Mean, Variance, Standard
Deviation, and Expectation
 Determining
the mean, standard
deviation, and variance of a probability
distribution is calculated differently from
samples.
 In
this section we will also look at another
measure known as expectation.
Section 5-3
Steps for the Mean of a Probability Distribution

Determine your probability distribution

Multiply each possible outcome by its
corresponding probability

Add the products up

The result is the mean of the probability distribution
Section 5-3
 Example
1:
Find the mean of the number of spots
that appear when a die is tossed.
Section 5-3
 Example
2:
In a family with two children, find the
mean of the number of children who will
be girls.
Section 5-3
 Example
3:
If three coins are tossed, find the
mean of the number of heads that occur.
Section 5-3
Steps for the Variance of a Probability Distribution

Determine your probability distribution

Determine your mean for the probability distribution

Multiply the square of each outcome by its
corresponding value, then sum up the products.

Subtract the square of the mean from the value you
calculated.

The result is the variance of the probability distribution
Section 5-3
Steps for the Standard Deviation of a Probability
Distribution

Determine your probability distribution

Determine your mean for the probability distribution

Determine your variance for the probability distribution

Take the square root of the variance.

The result is the standard deviation of the probability
distribution
Section 5-3
 Example
4:
Find the variance and standard
deviation for Example 1.
Section 5-3
 Example
5:
Five balls numbered 0, 2, 4, 6, and 8
are placed in a bag. After the balls are
mixed, one is selected, its number noted,
and replaced. If this experiment is
repeated many times, find the variance
and standard deviation of the numbers
on the balls.
Section 5-3
The expected value of a discrete variable
of probability distribution is the theoretical
average of the variable.
 The
symbol E(X) is used for expected value
 Expected
value is often used in games of
chance, insurance, etc.
 Expected
value is calculated in the same way
you would calculate theoretical mean.
Section 5-3
 Example
6:
One thousand tickets are sold at $1
each for a color television valued at $350.
What is the expected value of the gain if
a person purchases one ticket?
Section 5-3
 Example
7:
A financial advisor suggests that his
client select one of two types of bonds in
which to invest $5000. Bond X pays a
return of 4% and has a default rate of 2%.
Bond Y has a 2.5% return and default rate
of 1%. Find the expected rate of a return
and decide which bond would be a
better investment. When the bond
defaults, the investor loses all the
investment.
Section 4-5
Homework:
 Pg
253: 1 - 7