Math Tech IIII, Feb 26
Mean, Standard Deviation, and Expected Value
of Discrete Probability Distributions
Book Sections: 4.1
Essential Questions: How can I compute the probability of any event?
How do I compute and interpret mean, standard deviation and
expected value of a discrete probability distribution?
Standards: DA-5.10, DA-4.3, S.MD.1, .2, .3
Discrete Probability Distributions
• Discrete probability distribution – a table list all
possible values of a random variable and the
probability of that value occurring, and satisfying
the following conditions:
0 ≤ P(x) ≤ 1
 P(x)  1
Computing Statistics
• We can compute the mean and standard deviation
of a random variable, using the distribution table.
• This is a distribution and not a sample, we will
now call mean by another name, μ, and standard
deviation by σ (mu and sigma).
In here, we’re
just going to let
Texas handle this
end of it.
Discrete Probability Distributions
• In a discrete probability distribution the mean is
equal to the expected value of the distribution.
• Often, these terms are used interchangeably.
Calculator Computations
• We load all x values into L1 and all P(x) values in
L2, and do 1-var stat computations on both lists.
 Looks like: 1-Var Stats L1, L2
Using Calculator statistics
Name
Symbol
Mean
μ
Standard Deviation
σ
Look for on Calculator
X
σx
Example 1
Compute the mean and standard deviation of the following
discrete probability distribution:
x
0
P(x) 0.69
μ=
σ=
1
0.20
2
3
0.08 0.02
4
5
0.01 0.01
Example 2
Find the mean and standard deviation of the following discrete
probability distribution:
x
4
P(x) 0.31
μ=
σ=
6
0.10
8
10
0.22 0.07
12
15
0.18 0.12
Interpreting Statistics + 1
• The mean of a random variable is what you
would expect to happen over thousands of
trials
• The Expected value of a distribution is equal to
the mean of the distribution
• In a game situation, an expected value of 0
implies a fair game
• In a profit-loss analysis (or situation), an
expected value of 0 is the break even point
• A negative value gives you an expected loss
• A positive value gives you an expected gain
Creating a Distribution for a Situation
• To accomplish a gain-loss scenario, you can
create a probability distribution for the gains
and compute the mean. From this, you will
know expected gain (or loss if negative)
• Follow the next example, it is a template for all
such examples:
Example
At a raffle, 1500 tickets are sold at $2 each for four prizes of
$500, $250, $150, and $75. If you buy 1 ticket, what is your
expected gain. (Ans: The expected value of some distribution)
Example
At a raffle, 1500 tickets are sold at $2 each for four prizes of
$500, $250, $150, and $75. If you buy 1 ticket, what is your
expected gain. (Ans: The expected value of some distribution)
Here is the discrete probability distribution for this situation. We
will load it into the calculator, as is.
Gain, x
P(x)
$498
1
1500
$248 $148 $73
1
1500
1
1500
1
1500
-$2
1496
1500
μ=
Gain-Loss – Another Spin
• You can also compute the mean directly in a
gain-loss situation. This technique is especially
good if there are only 2 possible outcomes. It
will work for more, but gets more cumbersome.
• Expected value = x1·P(x1) + x2 ·P(x2)
Another Pivotal Example
A ball club makes $450,000 when it does not rain.
It also loses $250,000 when there is a rain out. If
the probability of it raining is .32, find the
expectation of profit?
Classwork: Handout CW 2/26, 1-6
Homework – None