Page 212 - Applied statistics and probability for engineers

190 Chapter 5/Joint Probability Distributions

Because the integral gives the probability that Y # a for all values of a contained in the feasible
⎡
set of values for y, f X ( )⎤ ⎦ ′(
u y u y) must be the probability density of Y. Therefore, the prob-
⎣
ability distribution of Y is
f Y ( y) = f X ( )⎤ ⎦ ′( f X ( )⎤ ⎦
u y u y) =
⎡
⎡
u y J
⎣
⎣
If the function y = h(x) is a decreasing function of x, a similar argument holds.
Example 5-35 Function of a Continuous Random Variable Let X be a continuous random variable with
probability distribution
f X ( x) = x , 0 ≤ x < 4
8
Find the probability distribution of Y = h(X) = 2X + 4.
Note that y = h(x) = 2x + 4 is an increasing function of x. The inverse solution is x = u(y) = (y – 4) / 2, and from this,

we ind the Jacobian to be J = u′(y) = dx / dy = 1 / 2. Therefore, from Equation 5-31, the probability distribution of Y is
⎛ ⎞
f Y ( y) = ( y − ) 4 / 2 1 = y − 4 , 4 ≤ y ≤ 12

⎜ ⎟
8 ⎝ ⎠ 2 32
EXERCISES FOR SECTION 5-5

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion

5-79. Suppose that X is a random variable with probabil- 5-85. Suppose that X has the probability distribution
ity distribution f X ( x) = 1 , 1 ≤ x ≤ 2
f X ( x) = 1 / , 4 x = 1 2 3 4 Determine the probability distribution of the random variable Y = e .
X
, , ,
Determine the probability distribution of Y = 2X + 1. 5-86. The random variable X has the probability distribution
5-80. Let X be a binomial random variable with p = 0.25 and f X ( x) = x , 0 ≤ x ≤ 4
n = 3. Determine the probability distribution of the random vari- 8
2
able Y = X . 2 Determine the probability distribution of Y = (X – 2) .
5-81. Suppose that X is a continuous random variable 5-87. An aircraft is l ying at a constant altitude with veloc-
with probability distribution ity magnitude r 1 (relative to the air) and angle θ 1 (in a two-
f X ( x) = x , 0 ≤ x ≤ 6 dimensional coordinate system). The magnitude and direction
18 of the wind are r 2 and θ 2 , respectively. Suppose that the wind
(a) Determine the probability distribution of the random vari- angle is uniformly distributed between 10 and 20 degrees and
able Y = 2X + 10. all other parameters are constant. Determine the probabil-
(b) Determine the expected value of Y. ity density function of the magnitude of the resultant vector
rr (cos 1 θ
. 0 5
5-82. Suppose that X has a uniform probability distribution r = [ 1 2 r + 1 2 − cosθ 2 )] .
2
r + 2
f X ( x) = 1 , 0 ≤ x ≤1 5-88. Derive the probability density function for a lognormal
Show that the probability distribution of the random variable Y random variable Y from the relationship that Y = exp( ) for a
W
= –2 X is chi-squared with two degrees of freedom. normal random variable W with mean θ and variance ω .
2
5-83. A random variable X has the probability distribution 5-89. The computational time of a statistical analysis applied
−
x
f X ( x) = e , x ≥ 0 to a data set can sometimes increase with the square of N, the
Determine the probability distribution for the following: number of rows of data. Suppose that for a particular algo-
2
2
(a) Y = X (b) Y = X / 1 2 (c) Y = ln X rithm, the computation time is approximately T = .0 004 N sec-
5-84. The velocity of a particle in a gas is a random vari- onds. Although the number of rows is a discrete measurement,
able V with probability distribution assume that the distribution of N over a number of data sets can
2
f v ( ) = av e − bv v > 0 be approximated with an exponential distribution with a mean
V
where b is a constant that depends on the temperature of the gas of 10,000 rows. Determine the probability density function and
and the mass of the particle. the mean of T .
(a) Determine the value of the constant a. 5-90. Power meters enable cyclists to obtain power meas-
(b) The kinetic energy of the particle is W = mV / 2. Determine urements nearly continuously. The meters also calculate the
2
the probability distribution of W. average power generated over a time interval. Professional