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(e)
From the formulas for the mean and variance of conditional narmal distribution we have
Reserve Problems Chapter 5 Section 5 Problem 6
Suppose that X and Y have a bivariate normal distribution with joint probability density function
( )
, ; , , , ,
XY x y x y
f x y
. Show that the conditional distribution of Y given that
Xx=
is
normal.
SOLUTION
By completing the square in the numerator of the exponent of the bivariate normal PDF, the joint
PDF can be written as
Reserve Problems Chapter 5 Section 5 Problem 7
Show that the probability density function
( )
, ; , , , ,
YY x y x y
f x y
of a bivariate normal
distribution integrates to 1.
[Hint: Complete the square in the exponent and use the fact that the integral of a normal
probability density function for a single variable is 1.]
SOLUTION
Let
Reserve Problems Chapter 5 Section 5 Problem 8
If X and Y have a bivariate normal distribution with joint probability density
( )
, ; , , , ,
XY x y x y
f x y
, show that the marginal probability distribution of X is normal with
mean
X
and standard deviation
X
.
[Hint: Complete the square in the exponent and use the fact that the integral of a normal
probability density function for a single variable is 1.]
SOLUTION
( )
( )
( )( )
22
22
2
2
( ) ( )
1
21
2
1d
21
XY
XY
XY
XY
xy
xy
X
XY
f x e y
−−
−−
−−+
−
−
= −
Reserve Problems Chapter 5 Section 6 Problem 1
Let X denote the active ingredient percentage (by weight) in a pharmaceutical capsule. It is
approximately normally distributed with a mean of 0.9% and a standard deviation of 0.02%. A
new mixing process is implemented and the mean percentage
X
of the active ingredient from a
sample of 23 independent capsules is 0.91%.
Determine
( )
0.91PX
Is it likely that the new mixing process also has a mean active ingredient percentage at 0.9%?
SOLUTION
Determine
( )
0.91PX
if X is approximately normally distributed with a mean of 0.9% and a
Reserve Problems Chapter 5 Section 6 Problem 2
The time for visitor to read health instructions on a Web site is approximately normally
distributed with a mean of 10 minutes and a standard deviation of 2 minutes. Suppose 64 visitors
independently view the site. Determine the following:
(a) Expected value and variance of the mean time of the visitors
(b) Probability that the mean time of the visitors is within 15 seconds of 10 minutes
(c) Value exceeded by the mean time of the visitors with probability 0.01.
SOLUTION
(a) Let X denote the time for visitor to read health instructions on a Web site.
The mean of X and
X
is
10
=
.
Reserve Problems Chapter 5 Section 6 Problem 3
Weights of parts are normally distributed with variance
2
. Measurement error is normally
distributed with mean 0 and variance
2
0.5
, independent of the part weights, and adds to the part
weight. Upper and lower specifications are at
3
about the process mean.
(a) Without measurement error, what is the probability that a part exceeds the specifications?
(b) With measurement error, what is the probability that a part is measured as being beyond
specifications? Does this imply it is truly beyond specifications?
(c) What is the probability that a part is measured as being beyond specifications if the true
weight of the part is
0.6
below the upper specification limit?
SOLUTION
(a)
W=
weights of a part and
E=
measurement error.
Reserve Problems Chapter 5 Section 6 Problem 4
Three electron emitters produce electron beams with changing kinetic energies that are
uniformly distributed in the ranges [2,11], [1,8], and [3,10]. Let Y denote the total kinetic energy
produced by these electron emitters.
(a) Suppose that the three beam energies are independent. Determine the mean and variance of Y.
(b) Suppose that the covariance between any two beam energies is –0.3. Determine the mean and
variance of Y.
(c) Compare and comment on the results in parts (a) and (b):
SOLUTION
(a)
1 2 3
Y X X X= + +
.
Reserve Problems Chapter 5 Section 7 Problem 1
Suppose that the length of a side of a cube X is uniformly distributed in the interval
9 10x
.
Determine the probability density function of the volume of the cube. Express your answers in
terms of V.
SOLUTION
Let X be the uniformly distributed random variable:
( ) 1
X
fx=
for
9 10x
.
Reserve Problems Chapter 5 Section 7 Problem 2
Let X be a binomial random variable with
0.2p=
and
3n=
. Determine the probability
distribution of the random variable
2
YX=
.
SOLUTION
Reserve Problems Chapter 5 Section 7 Problem 3
If Z is an exponential random variable with the parameter
,
1
2
n
i
i
Z
=
is a chi-square distributed
random variable with
2n
degrees of freedom. Suppose that X has a uniform probability
distribution
( )
1,0 1
X
f x x=
. Show that the probability distribution of the random variable
2lnYX=−
is chi-squared with two degrees of freedom.
SOLUTION
Reserve Problems Chapter 5 Section 7 Problem 4
The computational time of a statistical analysis applied to a data set can sometimes increase with
the square of N, the number of rows of data. Suppose that for a particular algorithm, the
computation time is approximately
2
0.004TN=
seconds. Although the number of rows is a
discrete measurement, assume that the distribution of N is over a number of data sets can be
approximated with an exponential distribution with a mean of 15000 rows. Determine the
probability density function and the mean of T.
SOLUTION
Reserve Supplemental Exercises Chapter 5 Problem 1
The random variables X and Y are uniformly distributed over the region
0x
,
0y
and
4xy+
. Determine the following:
(a)
22
( 2)P X Y+ =
(b)
( 2, 2)P X Y =
(c)
( )
EX =
(d)
( )
X
fx=
(e)
( )
|3Y
fy=
SOLUTION
Reserve Supplemental Exercises Chapter 5 Problem 2
Calculate the covariance and correlation between the random variables X and Y.
x
y
( )
,
XY
f x y
1.0
1
1/4
1.5
2
1/8
1.5
3
1/4
2.5
6
1/4
3.0
6
1/8
XY
=
XY
=
SOLUTION
Reserve Supplemental Exercises Chapter 5 Problem 3
Consider
( )
x
X
f x e−
=
,
0x
and joint probability density function
( )
,y
XY
f x y e−
=
for
0xy
.
Determine the following:
(a) Conditional probability distribution of Y given
1X=
.
(b)
( | 1)E Y X ==
(c)
( )
2 | 1P Y X = =
(d) Conditional probability distribution of X given
4Y=
.
SOLUTION
Reserve Supplemental Exercises Chapter 5 Problem 4
Consider the joint probability density function
( )
,f x y cxy=
for a constant c over the interior of
the region defined by the four lines
1, 1, 3 , 7y x y x y x y x= − = + = − = −
. Determine the
following:
(a)
c=
(b)
( 4)PX=
(c)
( 4)P X Y+ =
(d)
( )
EX =
(e)
( )
X
fx
(f)
( )
|3Y
fy
SOLUTION
(a) Let G denote the interior of the region defined by the four lines
Reserve Supplemental Exercises Chapter 5 Problem 5
The backoff torque required to remove bolts in a steel plate is rated as high, moderate, or low.
Historically, the probability of a high, moderate, or low rating is 0.6, 0.3, or 0.1, respectively.
Suppose that 20 bolts are evaluated and that the torque ratings are independent. Let X, Y, and Z
denote the number of bolts rated high, moderate, and low, correspondingly.
(a) What is the probability that 10, 8, and 2 bolts are rated as high, moderate, and low,
respectively?
(b) What is the marginal distribution of the number of bolts rated low?
(c) What is the expected number of bolts rated low?
(d) What is the probability that the number of bolts rated low is more than two?
(e) What is the conditional distribution of the number of bolts rated low given that 16 bolts are
rated high?
(f) What is the conditional expected number of bolts rated low given that 16 bolts are rated high?
(g) Are the numbers of bolts rated high and low independent random variables?
SOLUTION
(a)
