Applied Statistics and Probability for Engineers

5-7

chi-squared distribution with one degree of freedom. Let , and YZ^2. The

probability distribution of Zis the standard normal; that is,

The inverse solutions of yz^2 are so the transformation is not one to one. Define

and so that and. Then by Equation

S5-6, the probability distribution of Yis

Now it can be shown that , so we may write f(y) as

which is the chi-squared distribution with 1 degree of freedom.

EXERCISES FOR SECTION 5-8

fY 1 y 2 

1

21

2  a

1

2

b

y^1

2 ^1 ey^2 , y
0

1  11
22



1

21

21 

y^1

2 ^1 ey^2 , y
0

fY 1 y 2 

1

12 

ey^2 `

 1

21 y

`

1

12 

ey^2 `

1

21 y

`

z 1  1 y z 2  1 y J 1  11
22
1 y J 2  11
22
1 y

z 1 y,

f 1 z 2 

1

12 

ez

(^2
2)

,  z

Z 1 X 2


S5-1. Suppose that Xis a random variable with probability

distribution

Find the probability distribution of the random Y 2 X1.

S5-2. Let Xbe a binomial random variable with p0.25

and n3. Find the probability distribution of the random

variable YX^2.

S5-3. Suppose that Xis a continuous random variable with

probability distribution

(a) Find the probability distribution of the random variable

Y 2 X10.

(b) Find the expected value of Y.

S5-4. Suppose that Xhas a uniform probability distribution

Show that the probability distribution of the random variable

Y2 ln Xis chi-squared with two degrees of freedom.

fX 1 x 2 1, 0 x 1

fX 1 x 2 

x

18

, 0 x 6

fX 1 x 2  1
4, x1, 2, 3, 4

S5-5. A current of Iamperes flows through a resistance of R

ohms according to the probability distribution

Suppose that the resistance is also a random variable with

probability distribution

Assume that Iand Rare independent.

(a) Find the probability distribution for the power (in watts)

P I^2 R.

(b) Find E(P).

S5-6. A random variable Xhas the following probability

distribution:

(a) Find the probability distribution for YX^2.

(b) Find the probability distribution for Y.

(c) Find the probability distribution for Y ln X.

X 1

2

fX 1 x 2 ex, x 0

fR 1 r 2 1, 0 r 1

fI 1 i 2  2 i, 0 i 1

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