Page 219 - Applied Statistics And Probability For Engineers

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5-7

2
chi-squared distribution with one degree of freedom. Let Z 1X 2
, and Y Z . The
probability distribution of Z is the standard normal; that is,

1 2
f 1z2 e z
2 , z
12

2
The inverse solutions of y z are z 1y, so the transformation is not one to one. Define
z 1y and z 1y so that J 11
22
1y and J 11
22
1y . Then by Equation
1
2
1
2
S5-6, the probability distribution of Y is
1 1 1 1
f 1 y2 e y
2 ` ` e y
2 ` `
12 21y 12 21y
Y
1
1
2 y 1
2 1 y
2 , y
0
e
2 1
Now it can be shown that 1 11
22 , so we may write f(y) as
1
f 1 y2 y 1
2 1 y
2 , y
0
e
Y
1
1
2
2 a b
2
which is the chi-squared distribution with 1 degree of freedom.
EXERCISES FOR SECTION 5-8
S5-1. Suppose that X is a random variable with probability S5-5. A current of I amperes flows through a resistance of R
distribution ohms according to the probability distribution

f X 1x2 1
4, x 1, 2, 3, 4 f I 1i2 2i, 0 i 1

Find the probability distribution of the random Y 2X 1. Suppose that the resistance is also a random variable with
probability distribution
S5-2. Let X be a binomial random variable with p 0.25
and n 3. Find the probability distribution of the random
2
variable Y X . f R 1r2 1, 0 r 1
S5-3. Suppose that X is a continuous random variable with
probability distribution Assume that I and R are independent.
(a) Find the probability distribution for the power (in watts)
2
x P I R.
f X 1x2 , 0 x 6 (b) Find E(P).
18
S5-6. A random variable X has the following probability
distribution:
(a) Find the probability distribution of the random variable
Y 2X 10. x
(b) Find the expected value of Y. f X 1x2 e , x 0
S5-4. Suppose that X has a uniform probability distribution
2
(a) Find the probability distribution for Y X .
1
2 .
f X 1x2 1, 0 x 1 (b) Find the probability distribution for Y X
(c) Find the probability distribution for Y ln X.
Show that the probability distribution of the random variable
Y 2 ln X is chi-squared with two degrees of freedom.