MATH 141-501
Section 8.2 Lecture Notes
Measures of Central Tendency – give information about
the “center value” of a random variable
Mean
Mean: To obtain the mean (average) of a list of n numbers:
1. Add the numbers.
2. Divide by the sum from Step 1 by n.
Example: Compute the average of the four numbers 10, 12, 13, 9.
Median: To obtain the median of a list of n numbers:
1. Arrange the numbers in either increasing or decreasing order.
choice.)
(Your
2. If the number of entries is odd : the median is the middle number on the
list.
3. If the number of entries is even: the median is the average of the two
middle numbers on the list.
Example: Compute the median of the numbers 10, 12, 13, 9.
Example: Compute the median of the numbers 10, 7, 12, 13, 9
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Example: A small town has 10 residents. Four residents have net worths
of $0, three residents have a net worth of $1,000, and three residents have net
worths of $2,000, and five residents have net.
Find the mean and median net worth for this town.
One day a successful software engineer decides to move to this small town.
The software engineer has a net worth of $1,000,000. What is the mean and
median net worth after the engineer moves?
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Mode
The mode of a random variable (list of numbers) is the value which the
variable (list) takes most often.
1. If there is one number which occurs more than all the others, that number
is the mode.
2. If there is more than one number which occurs more than all the others,
those numbers are modes.
3. If all numbers occur equally often, there is no mode.
Example: Find the mode of the numbers 1, 1, 1, 2, 2, 3.
Example: Find the mode of the numbers 1, 1, 2, 2, 3
Example: Find the mode of the numbers 1, 1, 2, 2, 3, 3
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Averages With Large Numbers of Values
Example: The number of hits achieved by a major league baseball player
is recorded each night. The table is given below:
Hits
0
1
2
3
4
Number of Games
60
50
22
19
11
What is the average number of hits per night for this player?
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Expected Value
Expected Value (Formula): For a random variable X with probability
distribution given by
x
P (X = x)
x1
p1
···
···
x2
p2
xn
pn
the expected value E(X) is p1 x1 + p2 x2 + . . . + pn xn
Expected Value (Procedure): To find the value of a random variable X,
1. Multiply each possible value of X by its probability.
2. Add the numbers from Step 1 together.
Example: Find the expected value for the random variable X whose probability distribution is:
x
P (X = x)
2
0.15
5
4
0.6
6
0.25
Example - Insurance
A healthy, 35-year-old woman purchases a 10-year, $1,000,000 life insurance
policy for $200 a month.
(This means that the woman pays $200 a month for 10 years. If she dies during
the 1 year period, the company will pay the woman’s family $1,000,000. If she does
not die, the insurance company keeps her payments.)
Question: The insurance company estimates that there is a 0.1% chance
she will die during this 10-year-span. What is the expected value of the policy
for the insurance company?
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Example - Gambling
The gambling game of Pick-a-Card consists of picking a card from a well-shuffled
deck. The player bets $1. If the card is an ace or a two, the players wins $2.
Otherwise, the player loses the money. What is the expected value for the
player?
Definition: A game is fair if the expected value of the game is 0.
Question: Is Pick-a-Card a fair game?
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Odds
Often, the chance of an event is stated in terms of odds rather than probability. For instance, you might hear someone say:
“The odds of that happening are 3 to 1.” (i.e. the odds that it will
happen.)
OR:
“The odds against that happening are 5 to 2.” (i.e. the odds that it will
not happen.)
There are straightforward formulas that relate odds to probability, so that
we can convert back and forth:
Odds to Probability: If P (E) is the probability of an event E occuring,
then:
1. The odds in favor of E occuring are
P (E)
.
1 − P (E)
2. The odds against E occuring are
1 − P (E)
.
P (E)
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Example: In a game of No Limit Texas Hold ’Em poker, 12 cards have been
dealt and Roberto needs the Queen of Hearts to win the hand. (The Queen of
Hearts has not been dealt.) What are the odds in favor of Roberto winning
the hand?
Example: According to the National Weather Service, the probability of
being struck by lightning at some point in a person’s lifetime is 1/12,000. Find
the odds against being struck by lightning at some point in your lifetime.
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Converting Odds to Probability
If the odds in favor of an event E occurring are a to b, then the probability
of an event E occuring is
P (E) =
a
.
a+b
Example: At the beginning of the current NBA season, the website Vegas
Insider stated that the odds in favor of the San Antonio Spurs winning the NBA
title were 4-1. Find the probability (according to Vegas Insider) that the Spurs
will win the title this season.
If a person bets $100 on the Spurs to win the title, what is the expected
value of the bet?
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