Chapter 3: Probability Distributions and Statistics
Section 3.1-3.3
3.1 Random Variables and Histograms
A
is a rule that assigns precisely
one real number to each outcome of an experiment. We usually denote a random variable
by X.
Notes:
• The assignments of numbers to the outcomes of experiments are normally done in
a manner that is reasonable and, most importantly, in a manner that permits these
numbers to be used for interpretation and comparison.
• Unless otherwise specified, when the outcomes of an experiment are themselves
numbers, the random variable is the rule that assigns each number to itself.
Example 1 The following are examples of random variables:
1.) You flip a coin. Assign the random variable X the value 1 if heads and 2 if tails.
2.) You flip a coin 10 times. X is the number of heads that occur.
3.) You roll a die. X is the number that appears on the uppermost face of the die.
Types of Random Variables:
if it assumes only a
(1) A random variable is
finite number of values.
(2) A random variable is
if it takes
on an infinite number of values that can be listed in a sequence, so that there is a
first one, a second one, a third one, and so on.
(3) A random variable is said to be
take any of the infinite number of values in some interval of real numbers.
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if it can
Example 2 Classify the following random variables.
a) X = The number of defective batteries in a sample of 8
b) X = The number of times you flip a coin until a heads is shown
c) X = The amount of time an Aggie student stands during a home football game
A probability distribution of a random variable assigns a probability, p1 , p2 , ..., pn to
each value of the random variable, x1 , x2 , ..., xn . Recall that these can be represented by
probability distribution tables.
The probability distribution of a random variable X satisfies:
1. 0 ≤ pi ≤ 1
2. p1 + p2 + ... + pn = 1
Example 3 Determine whether the table gives the probability distribution of the random
variable X.
x
P (X = x)
-2 -1
0
1
2
0.1 0.2 0.2 0.3 0.1
Example 4 The probability distribution of the random variable X is shown in the table.
x
-5
-3
-2
0
2
3
P (X = x) 0.17 0.13 0.33 0.16 0.11 0.10
Find:
a) P (X = −3)
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b) P (X ≤ 2)
c) P (−3 ≤ X ≤ 10)
d) P (X > 0)
Example 5 Two fair four-sided dice are rolled. Let the random variable X denote the
sum of the faces that fall uppermost.
a) List the outcomes of the experiment.
b) Find the value assigned to each outcome of the experiment by the random variable X.
c) Find the probability distribution of X.
d) What is P (X ≥ 4)?
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e) What is P (3 < X ≤ 6)?
Example 6 A carton contains a dozen eggs, of which 5 are cracked. An experiment
consists of randomly selecting 3 eggs from the carton. Let X be the number of cracked
eggs that are selected.
a) Find the probability distribution for X.
b) What is the probability that more than 1 cracked egg will be selected?
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Example 7 An unfair die has the property that the probability of rolling a 1 is 0.3. This
die is rolled 4 times. Let X be the number of times a 1 is rolled. Find the probability
distribution for X.
Note: This is a binomial experiment (section 2.4). Therefore, this distribution is referred
to as a binomial distribution.
We can represent a probability distribution graphically using a
.
The x-axis of a histogram corresponds to the values of the random variable and the y-axis
corresponds to probabilities. Normally, we draw a bar of width one above each value of
the random variable and the height of the bar is the probability that the random variable
takes on that particular value.
Area and Probability: The
of a region of a histogram associated with
the random variable X is equal to P (X), the probability that X occurs. Furthermore,
the probability that X takes on the values in the range Xi ≤ X ≤ Xj is the sum of the
areas of the histogram from Xi to Xj .
Example 8 Draw a histogram for the probability distribution below. Shade the area
representing P (0 < X ≤ 3) and calculate its value.
X
0
1
2
3
4
5
Probability 0.2 0.15 0.3 0.1 0.05 0.2
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3.2 Measures of Central Tendency
The
by µ or x, is given by
or
of the n numbers x1 , x2 , ..., xn , denoted
Example 9 Find the mean of the following set of numbers: 64, 67, 68, 71
Example 10 Find the mean of the following set of numbers: 2, 64, 67, 68, 71
The mean is not always the best representation of what outcomes are occurring “most of
the time” because it is sensitive to extreme values. We will look at some other notions
of central tendency.
The
of a set of numerical data is the middle number when
the numbers are arranged in order of size and there is an odd number of entries in the
set. In the case that the number of entries in the set is even, the median is the mean of
the two middle numbers.
of a set of observations is the observation that occurs more frequently
The
than the others. If the frequency of occurence of two observations is the same and also
greater than the frequency of occurrence of all the other observations, then we say that the
set is bimodal and has two modes. If no one or two observations occurs more frequently
than the others, we say that the set has no mode.
Example 11 What are the mean, median, and mode of the set of numbers:
1, 4, 4, 5, 9, 9, 9? (Round your answers to four decimal places.)
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Example 12 What are the mean, median, and mode of the set of numbers:
5, 9, 4, 1, 5, 1? (Round your answers to four decimal places.)
Finding Mean and Median Using Your Calculator Given a List of Data:
(1) Press STAT.
(2) Select EDIT.
(3) In the first column, enter the values.
(4) Exit to the home screen.
(5) Press STAT.
(6) Scroll to the right to CALC.
(7) Select 1-Var Stats (Option 1)
(8) Press L1 (2nd → 1).
(9) Press ENTER.
(10) The mean is given by x. To find the median, scroll down until you see Med.
Example 13 The daily maximum temperature in degrees Fahrenheit for the month of
October in College Station follow:
85, 82, 85, 90, 89, 77, 63, 70, 85, 89,
89, 89, 86, 87, 82, 70, 90, 79, 82, 86,
90, 88, 86, 87, 88, 62, 63, 67, 71, 75, 84
Find the mean temperature and the median temperature. (Round to four decimal places.)
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Finding Mean and Median Using Your Calculator Given the Random Variable
and the Frequency or Probability:
(1) Press STAT.
(2) Select EDIT.
(3) In the first column, enter the values the random variable X takes on.
(4) In the second column, enter the frequency or probability for each value of X.
(5) Exit to the home screen.
(6) Press STAT.
(7) Scroll to the right to CALC.
(8) Select 1-Var Stats (Option 1)
(9) Press L1 (2nd → 1). Press the comma key. Press L2 (2nd → 2).
(10) Press ENTER.
(11) The mean is given by x. To find the median, scroll down until you see Med.
Example 14 A certain puzzle manufacturer makes and sells puzzles. The data below
shows how many puzzles this company makes with different numbers of pieces.
Number of Puzzles 400 200 350 150 275 300
75
Number of Pieces 500 1000 750 2000 300 1500 3000
a) Determine which row represents the random variable X and which row represents the
frequency.
b) Calculate the mean. (Round to four decimal places.)
Let X denote the random variable that has values x1 , x2 , ... xn , and let the associated
probabilities be p1 , p2 , ..., pn .
The
variable X, denoted by E(X) or µ, is
or
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of the random
Example 15 Calculate the mean of the above table using our new definition by hand.
(Round to four decimal places.)
Example 16 Find the expected value of a random variable X having the following probability distribution. Also find the mode(s).
X
Probability
2
3
4
5
6
1
6
1
3
1
4
1
12
1
6
5/12
1/3
1/4
1/6
1/12
0
2
3
4
X
9
5
6
Example 17 Two cards are selected at random from a standard deck of cards. What is
the expected number of hearts?
Expected values are often used in games to determine whether the game is “fair.” In a
game situation, we let X be the net winnings (profit) of the player. A game is considered
“fair” when the expected net winnings are 0, ie, when E(X) = 0.
Example 18 Visitors at a carnival pay $1.00 to play a game. The game consists of
pulling a marble out of a bag. If the marble is purple, the participant wins $5.00. If the
marble is maroon, the participant wins $2.00. If the marble is teal, the participant wins
$1.00. However, if the marble is white, the participant loses. Assume the bag contains 1
purple marble, 2 maroon marbles, 3 teal marbles, and 4 white marbles.
a) Draw a histogram of the probability distribution for this game where X represents your
net winnings.
b) Determine the expected winnings.
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c) Is this game fair?
Example 19 (From Tan) In a lottery, 5000 tickets are sold for $1 each. One first prize
of $2000, one second prize of $500, three third prizes of $100 and ten consolation prizes
of $25 are to be awarded. What are the expected net earnings of a person who buys a
ticket?
Example 20 A game consists of rolling a pair of fair 6-sided dice. The game costs $4
to play. If you roll the same number on both dice (a double), you win $a. Otherwise you
win nothing. What value of a would make this game fair?
Example 21 A life insurance company charges customers a yearly premium of $140 for
a $25,000 life insurance policy. If the probability that a given customer will live through
the next year is 0.998, what is the life insurance company’s net earnings?
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If the probability that the man lives drops to 0.98, what is the minimum amount of money
he can expect to pay for his policy?
The expected value of the binomial distribution with n trials and probability of success
p in a single trial and q = 1 − p is
Example 22 Given that a binomial trial is repeated 300 times with a probability of success
of 0.68, find the expected value of this binomial trial.
Example 23 Assume that the expected number of successes in a binomial experiment
consisting of 60 trials is 23. What is the probability of success?
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3.3 Measures of Spread
Example 24 The following lists are test scores for 10 students in Math 166. Calculate
the mean of each.
a) 44,48,55,56,58,95,97,98,99,100
b) 71,72,72,74,75,75,77,77,78,79
Goal: Measure the extent of the dispersion of the data from the mean.
Let X denote the random variable that takes on the values x1 , x2 , ..., xn , and let the
associated probabilities be p1 , p2 , ..., pn .
of the random variable, de-
Then if µ = E(X), the
noted by V ar(X), is
, denoted by σ(X) is
The
Example 25 Compare the variances of the following distributions.
A
B
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Finding Standard Deviation (and Variance) Using Your Calculator:
1. Press STAT, then EDIT
2. In the first column, enter the values the random variable X takes on.
3. In the second column, enter the frequency or probability for each value of X.
4. Exit to the home screen.
5. Press STAT. Scroll to the right to CALC. Select 1-Var Stats (Option 1). Press L1
(2nd → 1). Press the comma key. Press L2 (2nd → 2). Press ENTER.
6. The standard deviation is given by σx.
7. To find the variance, press VARS. Select Statistics... (Option 5). Select σx (Option
4). Press x2 . Press ENTER.
Example 26 Find the variance and standard deviation of the random variable X having
the following probability distribution.
X
Probability
2
3
4
5
6
1
6
1
3
1
4
1
12
1
6
Example 27 A certain puzzle manufacturer makes and sells puzzles. The data below
shows how many puzzles this company makes with different numbers of pieces.
Number of Pieces 500 1000 750 2000 300 1500 3000
Number of Puzzles 400 200 350 150 275 300
75
Find the standard deviation and variance for the number of pieces in a puzzle.
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