Unit 5
Continuous Random Variables
and Probability Distributions
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
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
X
X
X μZ
σ
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
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
Jointly Distributed Continuous Random Variables
X and Y are independent if and only if
(continued)
( , ) ( ) ( )F x y F x F y
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
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
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
Example – Bivariate Uniform
( , ) 1, 0 1, 0 1f x y x y
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
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)
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?
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?
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.
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
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
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
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
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
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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