Download AP Statistics Chapter 7 PowrerPoint

Daniel S. Yates
The Practice of Statistics
Third Edition
Chapter 7:
Random Variables
Copyright © 2008 by W. H. Freeman & Company
•
Discrete random variables
–
–
–
–
–
Number of sales
Number of calls
Shares of stock
People in line
Mistakes per page
•
Continuous random variables
–
–
–
–
–
Length
Depth
Volume
Time
Weight
Probability Histogram can be used to display the
probability distribution of a discrete random variable.
Probability distribution of digits in table of random digits
Probability distribution of the
random variable X, the number
of siblings of a randomly
selected student
Probability histogram for
the random variable X, the
number of siblings of a
randomly selected student
6
Probability Distribution of Grades
0=F
1=D
2=C
3=B
4=A
Ex. What is the probability distribution of the discrete
random variable X, that is the sum of a pair of dice.
X can take on values {2,3,4,5,6,7,8,9,10,11,12}
P(X=2) = 1/36 = 0.028
P(X=8) = 5/36 = 0.139
P(X=3) = 2/36 = 0.056
P(X=9) = 4/36 = 0.111
P(X=4) = 3/36 = 0.083
P(X=10) = 3/36 = 0.083
P(X=5) = 4/36 = 0.111
P(X=11) = 2/36 = 0.056
P(X=6) = 5/36 = 0.139
P(X=12) = 1/36 = 0.028
P(X=7) = 6/36 = 0.167
Ex. Probability
density curve
Probability Density Curves
• Any individual value has zero probability
• Only intervals between numbers have probability
• Therefore, P(X>a) and P(X>a) are equal
• Area under probability density curve equals 1
• The normal distribution is a probability density curve
Ex.
7.2 Means and Variances of random variables
aka. Expected value
Example
Find the mean (expected) size of an American household
Inhabitants
1
Proportion 0.25
of
households
2
3
4
5
6
7
0.32
0.17
0.15
0.07
0.03
0.01
mx = (1)(0.25) + (2)(0.32) + (3)(0.17) + (4)(0.15) +
(5)(0.07) +(6)(0.03) + (7)(0.01) = 2.6
Example
•
•
In a roulette wheel in a U.S.
casino, a $1 bet on “even” wins $1
if the ball falls on an even number
(same for “odd,” or “red,” or
“black”).
The odds of winning this bet are
47.37%
P( win $1) 18 / 38  .4737
P(lose $1)  20 / 38  .5263
m  $1 .4737  $1 .5263  .0526
On average, bettors lose about a nickel for each dollar they put down on a bet like this.
(These are the best bets for patrons.)
More About Means and Variances
• Adding or subtracting a constant from data
shifts the mean but doesn’t change the
variance or standard deviation:
m(x ± c) = m(x) ± c
s2(X ± c) = s2 (X)
– Example: Consider everyone in a company
receiving a $5000 increase in salary.
More About Means and Variances
(cont.)
• In general, multiplying each value of a
random variable by a constant multiplies the
mean by that constant and the variance by
the square of the constant:
2 2
2
m(ax) = am(x) s (ax) = a s (x)
– Example: Consider everyone in a company
receiving a 10% increase in salary.
More About Means and Variances
(cont.)
• In general,
– The mean of the sum of two random variables is the
sum of the means.
– The mean of the difference of two random variables is
the difference of the means.
m(x ± y) = m(x) ± m(y)
– If the random variables are independent, the variance
of their sum or difference is always the sum of the
variances.
s2(x ± y) = s2(x) + s2(y)
Example
Linda is a sales associate at a large auto dealership.
She estimates her car sales as follows:
Cars Sold
0
1
2
3
Probability
0.3
0.4
0.2
0.1
Calculate the mean (mx), variance (s2) and standard
deviation (s). X is the number of cars sold
xi
pi
x i pi
(xi-mx)2pi
0
0.3
0.0
(0-1.1)2(0.3) = 0.363
1
0.4
0.4
(1-1.1)2(0.4) = 0.004
2
0.2
0.4
(2-1.1)2(0.2) = 0.162
3
0.1
0.3
(3-1.1)2(0.1) = 0.361
mx = 1.1
s2x = 0.890
sx = 0.943
Linda also sells trucks and SUVs. Following is her
estimate of how many trucks and SUVs she will sell:
Trucks and
SUVs sold
0
1
2
Probability
0.4
0.5
0.1
Let Y be the number of trucks and SUVs she sells.
my = (0)(0.4) + (1)(0.5) + (2)(0.1) = 0.7 Trucks and SUVs
If she earns $350 for each car sold and $400 for each
truck or SUV. What are her expected earnings?
Let Z be her earnings; Z = 350X + 400Y
mz = 350mx + 400my = (350)(1.1) + (400)(0.7) = $665
What is the standard deviation of her earnings?
Trucks and
SUVs sold
0
1
2
Probability
0.4
0.5
0.1
(yi-my)2pi
(0-0.7)2(0.4)
= 0.196
(1-0.7)2(0.5)
= 0.045
(2-0.7)2(0.1)
= 0.169
s2y = 0.41
s z2  3502s x2  4002s y2
s  350 (0.890)  400 (0.41)
2
z
2
s z2  174625
s z  174625
2
s z  417.88