Download 7.1 Discrete and Continuous Random Variables

Daniel S. Yates
The Practice of Statistics
Third Edition
Chapter 7:
Random Variables
7.1 Discete and Continuous Random
Variables
Copyright © 2008 by W. H. Freeman & Company
1
AP Statistics, Section 7.1
Essential Questions
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What is discrete random variable?
What is a probability distribution?
How do you construct a probability distribution for a
discrete random variable?
Given a probability distribution for a random variable,
how do you construct a probability histogram?
What is a density curve?
What is a uniform distribution?
What is a continuous random variable and how do you
define a probability distribution for a continuous random
variable?
AP Statistics, Section 7.1
2
Random Variables
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A random variable is a variable whose value is
a numerical outcome of a random phenomenon.
For example: Flip three coins and let X
represent the number of heads. X is a random
variable.
We usually use capital letters to denotes
random variables.
The sample space S lists the possible values of
the random variable X.
We can use a table to show the probability
distribution of a discrete random variable.
AP Statistics, Section 7.1
3
Discrete Probability Distribution
Table
Value of X:
x1
x2
x3
…
xn
Probability:
p1
p2
p3
…
pn
AP Statistics, Section 7.1
4
Discrete Random Variables
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A discrete random variable X has a
countable number of possible values. The
probability distribution of X lists the
values and their probabilities.
X:
x 1 x 2 x 3 … xk
P(X): p1 p2 p3 … pk
1. 0 ≤ pi ≤ 1
2. p1 + p2 + p3 +… + pk = 1.
AP Statistics, Section 7.1
5
Probability Distribution Table:
Number of Heads Flipping 4 Coins
TTTT
TTTH
TTHT
THTT
HTTT
TTHH
THTH
HTTH
HTHT
THHT
HHTT
THHH
HTHH
HHTH
HHHT
HHHH
X
0
1
2
3
4
P(X)
1/16
4/16
6/16
4/16
1/16
AP Statistics, Section 7.1
6
Probabilities:
X:
 P(X):
0
1
1/16 1/4
.0625 .25
 Histogram
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2
3/8
.375
AP Statistics, Section 7.1
3
1/4
.25
4
1/16
.0625
7
Questions.
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Using the previous probability distribution for the
discrete random variable X that counts for the
number of heads in four tosses of a coin. What
are the probabilities for the following?
P(X = 2)
.375
.375 + .25 + .0625 = .6875
P(X ≥ 2)
1-.0625 = .9375
P(X ≥ 1)
AP Statistics, Section 7.1
8
What is the average number of
heads?
x  0   1  2   3   4 
1
16
6
16
4
16

0
16


32
16
2
4
16

12
16

12
16

4
16
1
16
4
16
AP Statistics, Section 7.1
9
Continuous Random Varibles
Suppose we were to randomly generate a
decimal number between 0 and 1. There are
infinitely many possible outcomes so we clearly
do not have a discrete random variable.
 How could we make a probability distribution?
 We will use a density curve, and the probability
that an event occurs will be in terms of area.
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AP Statistics, Section 7.1
10
Distribution of Continuous Random
Variable
AP Statistics, Section 7.1
11
Problem
Let X be the amount of time (in minutes)
that a particular San Francisco commuter
must wait for a BART train. Suppose that
the density curve is a uniform distribution.
 Draw the density curve for 0 to 20
minutes.
 What is the probability that the wait is
between 12 and 20 minutes?
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AP Statistics, Section 7.1
12
Density Curve.
Distribution Plot
Uniform, Lower=0, Upper=20
0.05
Density
0.04
0.03
0.02
0.01
0.00
0
5
10
X Section 7.1
AP Statistics,
15
20
13
Probability shaded.
Distribution Plot
Uniform, Lower=0, Upper=20
0.4
0.05
Density
0.04
0.03
0.02
0.01
0.00
0
X
12
20
P(12≤ X ≤ 20) = 0.5 · 8 = .40
AP Statistics, Section 7.1
14
Normal Curves
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We’ve studied a density curve for a continuous
random variable before with the normal distribution.
Recall: N(μ, σ) is the normal curve with mean μ and
standard deviation σ.
If X is a random variable with distribution N(μ, σ),
then Z  X  
is N(0, 1)

AP Statistics, Section 7.1
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Example
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Students are reluctant to report cheating by
other students. A sample survey puts this
question to an SRS of 400 undergraduates: “You
witness two students cheating on a quiz. Do you
go to the professor and report the cheating?”
p̂
Suppose that if we could ask all undergraduates,
12% would answer “Yes.” The proportion p =
0.12 would be a parameter for the population of
all undergraduates.
The proportion pˆ of the sample who answer " yes" is a statistic
used to estimate p. pˆ is a random variable with a distributi on of N(0.12, 0.016).
AP Statistics, Section 7.1
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Example continued
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Students are reluctant to report cheating by
other students. A sample survey puts this
question to an SRS of 400 undergraduates: “You
witness two students cheating on a quiz. Do you
go to the professor and report the cheating?”
What is the probability that the survey results
differs from the truth about the population by
more than 2 percentage points? pˆ  0.10 or pˆ  0.14.
Because p = 0.12, the survey misses by more
than 2 percentage points if pˆ  0.10 or pˆ  0.14.
AP Statistics, Section 7.1
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AP Statistics, Section 7.1
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Example continued Calculations
P( pˆ  0.10 or pˆ  0.14)  1  P(0.10  pˆ  0.14)
From Table A,
 0.10  0.12 pˆ  0.12 0.14  0.12 
P(0.10  pˆ  0.14)  P 



0.016
0.016
0.016


 P(1.25  Z  1.25)
 0.8944  0.1056  0.7888
So,
P( pˆ  0.10 or pˆ  0.14)  1  0.7888  0.2112
About 21% of sample results will be off by more than two
percentage points.
AP Statistics, Section 7.1
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Summary
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A discrete random variable X has a countable
number of possible values.
The probability distribution of X lists the
values and their probabilities.
A continuous random variable X takes all
values in an interval of numbers.
The probability distribution of X is described
by a density curve. The Probability of any event
is the area under the density curve and above
the values of X that make up the event.
AP Statistics, Section 7.1
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Summary
When you work problems, first identify the
variable of interest.
 X = number of _____ for discrete random
variables.
 X = amount of _____ for continuous
random variables.
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AP Statistics, Section 7.1
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