Download Random Variable

Chapter 12
Probability and Calculus
© 2010 Pearson Education Inc.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 1 of 15
Chapter Outline

Discrete Random Variables

Continuous Random Variables

Expected Value and Variance

Exponential and Normal Random Variables

Poisson and Geometric Random Variables
© 2010 Pearson Education Inc.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 2 of 15
§ 12.1
Discrete Random Variables
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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 3 of 15
Section Outline

Mean

Expected Value

Variance

Standard Deviation

Frequency Table

Relative Frequency Table

Relative Frequency Histogram

Random Variable

Applications of Expected Value, Variance, and Standard Deviation
© 2010 Pearson Education Inc.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 4 of 15
Mean
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Definition
Example
Mean: The sum of a
set of numbers,
divided by how many
numbers were
summed
2,3,3,5,8,12
2  3  3  5  8  12
6
 5.5
mean 
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 5 of 15
Expected Value
• a1 is the first number from a set of numbers, a2 is the
second and so on
• p1 is the probability that a1 occurs, p2 is the
probability that a2 occurs and so on
© 2010 Pearson Education Inc.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 6 of 15
Variance
• m is the expected value (or mean) of the set of
numbers
• a1 is the first number from a set of numbers, a2 is the
second and so on
• p1 is the probability that a1 occurs, p2 is the
probability that a2 occurs and so on
© 2010 Pearson Education Inc.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 7 of 15
Standard Deviation
standard
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deviation   variance
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 8 of 15
Frequency Table (Distribution)
Definition
Example
Frequency Table: A list
containing a set of numbers
and the frequency with which
each occurs
© 2010 Pearson Education Inc.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 9 of 15
Relative Frequency Table (Probability Table)
Definition
Example
Relative Frequency Table: A
list containing a set of
numbers and the relative
frequency (percent of the
time) with which each occurs
© 2010 Pearson Education Inc.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 10 of 15
Relative Frequency Histogram
Definition
Example
Relative Frequency
Histogram: A graph
where over each grade
we place a rectangle
whose height equals the
relative frequency of that
grade
(Compare with relative frequency
table from last slide)
© 2010 Pearson Education Inc.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 11 of 15
Random Variable
Definition
Example
Random Variable: A variable
whose value depends entirely
on chance
Examples will follow.
© 2010 Pearson Education Inc.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 12 of 15
Applications of Expected Value, Variance, & Standard Deviation
EXAMPLE
Find E(X), Var (X), and the standard deviation of X, where X is the random
variable whose probability table is given in Table 5.
SOLUTION
4
4
1
E X   1   2   3   1.7
9
9
9
2 4
2 4
2 1
Var  X   1  1.7   2  1.7   3  1.7   0.45
9
9
9
Standard Deviation  X   0.45  0.67
© 2010 Pearson Education Inc.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 13 of 15
Applications of Expected Value
EXAMPLE
The number of phone calls coming into a telephone switchboard during each
minute was recorded during an entire hour. During 30 of the 1-minute
intervals there were no calls, during 20 intervals there was one call, and during
10 intervals there were two calls. A 1-minute interval is to be selected at
random and the number of calls noted. Let X be the outcome. Then X is a
random variable taking on the values 0, 1, and 2.
(a) Write out a probability table for X.
(b) Compute E(X).
(c) Interpret E(X).
SOLUTION
(a)
Number of Calls
0
1
2
Probability 30/60 = 1/2 20/60 = 1/3 10/60 = 1/6
© 2010 Pearson Education Inc.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 14 of 15
Applications of Expected Value
CONTINUED
(b)
1
1
1
E X   0   1   2   0.67
2
3
6
(c) Since E(X) = 0.67, this means that the expected number of phone calls in a
1-minute period is 0.67 phone calls.
© 2010 Pearson Education Inc.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 15 of 15