Page 219 - Applied statistics and probability for engineers

Section 5-6/Moment-Generating Functions 197

5-115. Suppose that X and Y have a bivariate normal distribu- graphics enhancement only, extra memory only, and both,
tion with σ = 4, σ = 1, /, μ = 4, and ρ = − .0 2. Draw a rough respectively.
Y
Y
X
contour plot of the joint probability density function. (a) Describe the range of the joint probability distribution of X,
Y, and Z.
5-116. If XY ( ) = 1 exp ⎧ ⎨ −1 ⎡ ( x − ) 1 2 (b) Is the probability distribution of X, Y, and Z multinomial?
x, y
f
. π
1 2 ⎩ . 0 72 ⎣ ⎢ Why or why not?
⎤
2
1
− . ( 6 x − )( y − ) + ( y − ) 2 2 2 ⎦ ⎥} (c) Determine the conditional probability distribution of X
1
given that Y = 2.
determine E(X), E(Y), V(X), V(Y), and ρ by reorganizing the Determine the following:
(
(
parameters in the joint probability density function. (d) P X = 1 ,Y = 2 ,Z = ) 1 (e) P X = 1 ,Y = ) 1
(
5-117. The permeability of a membrane used as a (f) E X ( ) and V X ( ) (g) P X = 1 ,Y = Z | 2 = ) 1
(
moisture barrier in a biological application depends on the
(h) P X = |2 Y = ) 2
thickness of two integrated layers. The layers are normally
(i) Conditional probability distribution of X given that Y = 0
distributed with means of 0.5 and 1 millimeters, respec-
and Z = 3.
tively. The standard deviations of layer thickness are 0.1
and 0.2 millimeters, respectively. The correlation between 5-121. A marketing company performed a risk analysis
layers is 0.7. for a manufacturer of synthetic ibers and concluded that new
(a) Determine the mean and variance of the total thickness of competitors present no risk 13% of the time (due mostly to
the two layers. the diversity of ibers manufactured), moderate risk 72% of
(b) What is the probability that the total thickness is less than 1 the time (some overlapping of products), and very high risk
millimeter? (competitor manufactures the exact same products) 15% of the
(c) Let X and X denote the thickness of layers 1 and 2, respec- time. It is known that 12 international companies are planning
1 2
tively. A measure of performance of the membrane is a to open new facilities for the manufacture of synthetic ibers
function of 2X + 3X of the thickness. Determine the mean within the next three years. Assume that the companies are
1 2
and variance of this performance measure. independent. Let X, Y, and Z denote the number of new com-
5-118. The permeability of a membrane used as a mois- petitors that will pose no, moderate, and very high risk for the
ture barrier in a biological application depends on the thick- interested company, respectively.
ness of three integrated layers. Layers 1, 2, and 3 are normally Determine the following:
distributed with means of 0.5, 1, and 1.5 millimeters, respec- (a) Range of the joint probability distribution of X, Y, and Z
(
(
tively. The standard deviations of layer thickness are 0.1, 0.2, (b) P X = 1, Y = 3, Z = ) 1 (c) P Z ≤ ) 2
(
and 0.3, respectively. Also, the correlation between layers 1 ( = ) (e) P Z ≤ | X = )
(d) P Z = |2 Y = 1 , X 10 1 10
| (
(
and 2 is 0.7, between layers 2 and 3 is 0.5, and between layers X = ) (g) E Z X = )
(f) P Y ≤1 ,Z ≤ | 10 10
1
1 and 3 is 0.3.
(a) Determine the mean and variance of the total thickness of 5-122. Suppose X has a lognormal distribution with param-
the three layers. eters θ and ω. Determine the probability density function and
γ
(b) What is the probability that the total thickness is less than the parameters values for Y = X for a constant γ > 0. What is
1.5 millimeters? the name of this distribution?
/
2
5-119. A small company is to decide what investments to 5-123. The power in a DC circuit is P = I R where I and R
use for cash generated from operations. Each investment has denote the current and resistance, respectively. Suppose that
a mean and standard deviation associated with the percent- I is normally distributed with mean of 200 mA and standard
age gain. The irst security has a mean percentage gain of deviation 0.2 mA and R is a constant. Determine the probabil-
5% with a standard deviation of 2%, and the second security ity density function of power.
2
provides the same mean of 5% with a standard deviation 5-124. The intensity (mW/mm ) of a laser beam on a surface
of 4%. The securities have a correlation of –0.5, so there theoretically follows a bivariate normal distribution with maxi-
is a negative correlation between the percentage returns. If mum intensity at the center, equal variance σ in the x and y
the company invests two million dollars with half in each directions, and zero covariance. There are several deinitions
security, what are the mean and standard deviation of the for the width of the beam. One deinition is the diameter at
percentage return? Compare the standard deviation of this which the intensity is 50% of its peak. Suppose that the beam
strategy to one that invests the two million dollars into the width is 1.6 mm under this deinition. Determine σ. Also deter-
irst security only. mine the beam width when it is deined as the diameter where
2
5-120. An order of 15 printers contains 4 with a graph- the intensity equals 1/ e of the peak.
ics-enhancement feature, 5 with extra memory, and 6 with 5-125. Use moment generating functions to determine the
/
1 4
both features. Four printers are selected at random, with- normalized power [ (E X 4 )] from a cycling power meter when
out replacement, from this set. Let the random variables X, X has a normal distribution with mean 200 and standard devia-
Y, and Z denote the number of printers in the sample with tion 20 Watts.