statistics for business and economics, 7/e · let x 1, x 2, . . ., x k be continuous random...
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Unit 5
Continuous Random Variables
and Probability Distributions
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Linear Functions of Random Variables
Let W = a + bX , where X has mean μX and
variance σX2 , and a and b are constants
Then the mean of W is
the variance is
the standard deviation of W is
XW bμabX]E[aμ
2
X
22
W σbbX]Var[aσ
XW σbσ
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Example
An important special case of the result for the linear
function of random variable is the standardized random
variable
Find the mean and standard deviation of Z
X
X
X μZ
σ
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X
X
X μZ
σ
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Jointly Distributed Continuous Random Variables
Let X and Y be continuous random variables
Their joint cumulative distribution function, is defined
as
0 0
0 0 0 0( , ) ( ) ( , )
y x
F x y P X x Y y f x y dxdy
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Jointly Distributed Continuous Random Variables
The cumulative distribution functions
F(x) and F(y)
of the individual random variables are called their
marginal distribution functions
(continued)
( ) ( , )F x f x y dy
( ) ( , )F y f x y dx
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Jointly Distributed Continuous Random Variables
X and Y are independent if and only if
(continued)
( , ) ( ) ( )F x y F x F y
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Covariance
Let X and Y be jointly distributed continuous random
variables, with means μx and μy
The expected value of (X - μx)(Y - μy) is called the
covariance between X and Y
)]μ)(YμE[(XY)Cov(X, yx
( )( ) ( , )X Y
x y f x y dxdy
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Covariance
If X and Y are independent, then the covariance
between them is 0.
However, the converse is not true EXCEPT in the case
when the jointly normally distributed
Covariance measures linear dependence only
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Correlation
Let X and Y be jointly distributed continuous random
variables standard deviations sx and sy
The correlation between X and Y is
YXσσ
Y)Cov(X,Y)Corr(X,ρ
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Example – Bivariate Uniform
( , ) 1, 0 1, 0 1f x y x y
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Linear Combinations of Random Variables
A linear combination of two random variables, X and Y, (where a and b are constants) is
The mean of W is
bYaXW
YXW bμaμbY]E[aXE[W]μ
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Linear Combinations of Random Variables
The variance of W is
Or using the correlation,
If both X and Y are joint normally distributed random variables then the linear combination, W, is also normally distributed
Y)2abCov(X,σbσaσ 2
Y
22
X
22
W
YX
2
Y
22
X
22
W σY)σρ(X,2abσbσaσ
(continued)
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Example
Two tasks must be performed by the same worker.
X = minutes to complete task 1; μx = 20, σx = 5
Y = minutes to complete task 2; μy = 20, σy = 5
X and Y are normally distributed and independent
What is the mean and standard deviation of the time to
complete both tasks?
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Example
Two tasks must be performed by the same worker.
X = minutes to complete task 1; μx = 20, σx = 5
Y = minutes to complete task 2; μy = 20, σy = 5
X and Y are normally distributed and independent
What is the mean and standard deviation of the difference
in time to complete both tasks?
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Example
Two tasks must be performed by the same worker.
X = minutes to complete task 1; μx = 20, σx = 5
Y = minutes to complete task 2; μy = 20, σy = 5
X and Y are normally distributed and independent
Find the probability that task1 takes longer than task 2.
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Jointly Distributed Continuous Random Variables
Let X1, X2, . . ., Xk be continuous random variables
Their joint cumulative distribution function,
F(x1, x2, . . ., xk)
defines the probability that simultaneously X1 is less
than x1, X2 is less than x2, and so on; that is
)xXxXxP(X)x,,x,F(x kk2211k21
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Jointly Distributed Continuous Random Variables
The cumulative distribution functions
F(x1), F(x2), . . ., F(xk)
of the individual random variables are called their
marginal distribution functions
The random variables are independent if and only if
(continued)
)F(x))F(xF(x)x,,x,F(x k21k21
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Sums of Random Variables
1 1 2 2Y k ka a a
1 2
1 1 2 2 k
1 2
Let , , , be random variables
with means , 1 .
Let X
for constants , , , .
Then the mean of Y is
k
i
k
k
X X X k
i k
Y a X a X a
a a a
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Sums of Random Variables
2 2 2 2 2 2 2
1 1 2 2Y k ka a as s s s
1 2
2
i
1 1 2 2 k
1 2
Let , , , be random variables
with variances , 1 .
Let X
for constants , , , .
Then the variance of Y is
k
k
k
X X X k
i k
Y a X a X a
a a a
s
If the covariance between every pair of these random variables is 0, then the variance of their sum is the sum of their variances
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Sums of Random Variables
2 2 2 2 2 2 2
1 1 2 22 ( , )
Y k k i j i ji j
a a a a a Cov X Xs s s s
1 2
2
i
1 1 2 2 k
1 2
Let , , , be random variables
with variances , 1 .
Let X
for constants , , , .
Then the variance of Y is
k
k
k
X X X k
i k
Y a X a X a
a a a
s
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Example
It is estimated that in normal highway driving the
number of miles that can be covered by
automobiles of a particular model on 1 gallon of
gasoline is a normally distributed random variable
with mean 28 mpg and standard deviation 2.4 mpg.
Four of these cars, each with 1 gallon of gasoline,
are driven independently under highway conditions.
Find the probability that the average mpg of these
four cars exceeds 30 mpg.
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