Download random variable x

Section 5.1
Review and Preview
Review and Preview
This chapter combines the methods of descriptive
statistics presented in Chapter 2 and 3 and those of
probability presented in Chapter 4 to describe and
analyze probability distributions.
Probability Distributions describe what will probably
happen instead of what actually did happen, and
they are often given in the format of a graph, table,
or formula.
Preview
In order to fully understand probability distributions,
we must first understand the concept of a random
variable, and be able to distinguish between discrete
and continuous random variables. In this chapter we
focus on discrete probability distributions. In
particular, we discuss binomial probability
distributions.
Combining Descriptive Methods
and Probabilities
In this chapter we will construct probability distributions by
presenting possible outcomes along with the relative frequencies
we expect.
Section 5.2
Random Variables
Random Variable
Probability Distribution
Random variable: a variable (typically represented
by x) that has a single numerical value, determined
by chance, for each outcome of a procedure.
EX: the number of peas with green pods among 5 offspring peas
Probability distribution: a description that gives the
probability for each value of the random variable;
often expressed in the format of a graph, table, or
formula.
Discrete and Continuous Random
Variables
Discrete random variable: either a finite number of
values or countable number of values, where
“countable” refers to the fact that there might be
infinitely many values, but they result from a counting
process. *cannot be a decimal!!!
Continuous random variable: infinitely many values, and
those values can be associated with measurements on a
continuous scale without gaps or interruptions.
*could be a decimal!!!
Example 1: Identify the given random variable as being
discrete or continuous.
a) The number of people now driving a car in the
United States.
b) The weight of the gold stored in Fort Knox.
c) The height of the last airplane departed from JFK
Airport in New York City.
Example 1 continued: Identify the given random
variable as being discrete or continuous.
d) The number of cars in San Francisco that crashed last
year.
e) The time required to fly from Los Angeles to
Shanghai.
Requirements for
Probability Distribution
The sum of all probabilities is 1.
ΣP(x) = 1, where x assumes all possible values.
(values such as 0.999 or 1.001 are acceptable because they result from rounding errors)
Each individual probability is a value between 0 and
1 inclusive.
0  P(x)  1, for every individual value of x.
Formulas for a
Probability Distribution
µ = Σ [x • P(x)]
Mean
2
σ=
Σ[x • P(x)] – µ
2
2
2
σ = Σ[(x – µ) • P(x)]
2
2
σ = Σ[x • P(x)] – µ
2
Standard Deviation
Variance
Variance (shortcut)
Roundoff Rule for µ, , and 
2
Round results by carrying one more decimal place
than the number of decimal places used for the
random variable x.
If the values of x are integers, round µ, σ, and σ2 to
one decimal place.
Example 2: Determine whether or not a probability distribution is
given. If a probability distribution is given, find its mean and
standard deviation. If a probability distribution is not given, identify
the requirement(s) that are not satisfied. Three males with X-linked
genetic disorder have one child each. The random variable x is the
number of children among the three who inherit the X-linked genetic
disorder.
𝟐
𝟐
x
P(x)
0
0.125
1
0.375
2
0.375
3
0.125
𝒙 ∙ 𝑷(𝒙)
𝒙
𝒙 ∙ 𝑷(𝒙)
Example 3: Determine whether or not a probability distribution is given. If a
probability distribution is given, find its mean and standard deviation. If a
probability distribution is not given, identify the requirement(s) that are not
satisfied. Air America has a policy of routinely overbooking flights. The random
variable x represents the number of passengers who cannot be boarded because
there are more passengers than seats (based on data from an IBM research paper
by Lawrence, Hong, and Cherrier.)
x
P(x)
0
0.051
1
0.141
2
0.274
3
0.331
4
0.187
Example 4: Determine whether or not a probability
distribution is given. If a probability distribution is
given, find its mean and standard deviation. If a
probability distribution is not given, identify the
requirement(s) that are not satisfied.
x
P(x)
1
0.6
2
0.2
3
0.2
4
0.15
5
0.05
Identifying Unusual Results
Range Rule of Thumb
According to the range rule of thumb, most
values should lie within 2 standard deviations of
the mean.
We can therefore identify “unusual” values by
determining if they lie outside these limits:
Maximum usual value = μ + 2σ
Minimum usual value = μ – 2σ
Identifying Unusual Results
Probabilities
Rare Event Rule for Inferential Statistics
If, under a given assumption (such as the
assumption that a coin is fair), the probability of
a particular observed event (such as 992 heads in
1000 tosses of a coin) is extremely small, we
conclude that the assumption is probably not
correct.
Identifying Unusual Results
Probabilities
Using Probabilities to Determine When Results
Are Unusual
Unusually high: x successes among n trials is an
unusually high number of successes if
P(x or more) ≤ 0.05.
Unusually low: x successes among n trials is an
unusually low number of successes if
P(x or fewer) ≤ 0.05.
Example 5: Refer to the table, which describes results from eight
offspring peas. The random variable x represents the number of
offspring peas with green pods.
a) Find the probability of getting exactly
7 peas with green pods.
b) Find the probability of getting 7 or
more peas with green pods.
c) Which probability is relevant for
determining whether 7 is an unusually
high number of peas with green pods: the
result from part (a) or part (b)?
d) Is 7 an unusually high number of peas
with green pods? Why or why not?
x
P(x)
0
0+
1
0+
2
0.004
3
0.023
4
0.087
5
0.208
6
0.311
7
0.267
8
0.100
Example 6: Based on past results found in the Information Please
Almanac, there is a 0.1919 probability that a baseball World Series
contest will last four games, a 0.2121 probability that it will last five
games, a 0.2222 probability that it will last six games, and a 0.3737
probability that it will last seven games.
a) Does the given information describe a probability distribution?
c) Is it unusual for a team to “sweep” by winning in four games? Why or why not?
b) Assuming that the given information describes a probability distribution, find the mean
and standard deviation for the numbers of games in a World Series contest.
x
4
5
6
7
P(x)
0.1919
0.2121
0.2222
0.3737
x
P(x)
4
0.1919
5
0.2121
6
0.2222
7
0.3737
𝒙 ∙ 𝑷(𝒙)
𝒙 ∙ 𝑷(𝒙)
𝒙𝟐
𝒙𝟐
𝒙𝟐 ∙ 𝑷(𝒙)
𝒙𝟐 ∙ 𝑷(𝒙)
Example 8
Let the random variable x represent the number of girls in a family of three
children. Construct a table describing the probability distribution, then find the
mean and standard deviation. (HINT: List the different possible outcomes.)
Is it unusual for a family of three children to consist of three girls?
Different possible outcomes of having three children:
0 girls: BBB
1 girl: GBB
2 girls: GGB
3 girls: GGG
x
Mean:
BGB
BBG
GBG
BGG
P(x)
Standard Deviation:
𝒙 ∙ 𝑷(𝒙)
𝒙𝟐
0
1
2
3
Is it unusual for a family of three children to consist of three girls?
𝒙𝟐 ∙ 𝑷(𝒙)
Expected Value
The expected value of a discrete random
variable is denoted by E, and it represents the
mean value of the outcomes. It is obtained by
finding the value of Σ [x • P(x)].
E = Σ[x • P(x)]
Example 9: In the Illinois Pick 3 lottery game, you pay
50¢ to select a sequence of three digits, such as 233. If
you select the same sequence of three digits that are
drawn, you win and collect $250.
a) How many different selections are possible?
b) What is the probability of winning?
c) If you win, what is your net profit?
Example 9 continued: In the Illinois Pick 3 lottery
game, you pay 50¢ to select a sequence of three digits,
such as 233. If you select the same sequence of three
digits that are drawn, you win and collect $250.
d) Find the expected value.
e) If you bet 50 ¢ in Illinois’ Pick 4 game, the expected
value is –25¢. Which bet is better: A 50¢ bet in the
Illinois Pick 3 game or a 50¢ bet in the Illinois Pick 4
game? Explain.
Example 10: When playing roulette at the Bellagio casino in
Las Vegas, a gambler is trying to decide whether to bet $5
on the number 13 or to bet $5 that the outcome is any one
of these five possibilities: 0 or 00 or 1 or 2 or 3. The
expected value for a $5 bet for a single number is –26¢.
For the $5 bet that the outcome is 0 or 00 or 1 or 2 or 3,
there is a probability of 5/38 of making a net profit of $30
and a 33/38 probability of losing $5.
a) Find the expected value for the $5 bet that the outcome
is 0 or 00 or 1 or 2 or 3.
Example 10 continued: When playing roulette at the
Bellagio casino in Las Vegas, a gambler is trying to decide
whether to bet $5 on the number 13 or to bet $5 that the
outcome is any one of these five possibilities: 0 or 00 or 1
or 2 or 3. The expected value for a $5 bet for a single
number is –26¢. For the $5 bet that the outcome is 0 or 00
or 1 or 2 or 3, there is a probability of 5/38 of making a net
profit of $30 and a 33/38 probability of losing $5.
b) Which bet is better: A $5 bet on the number 13 or a $5
bet that the outcome is 0 or 00 or 1 or 2 or 3? Why?