Chapter 4 Random Variables and Expectation 137 6. The amount of time, in hours, that a computer functions before breaking down is a continuous random variable with probability density function given by de/100 f(x)%3D What is the probability that a computer will function between 50 and 150 hours before breaking down? What is the probability that it will function less than 100 hours? 7. The lifetime in hours of a certain kind of radio tube is a random variable having a probability density function given by x 100 f (x) = 100 *> 100 What is the probability that exactly 2 of 5 such tubes in a radio set will have to be replaced within the first 150 hours of operation? Assume that the events E, i= 1, 2, 3, 4, 5, that the ith such tube will have to be replaced within thi time are independent. 8. If the density function of X equals Ice-2 0 x<00 f(x) = { x <0 find c. What is P{X > 2}? 9. A set of five transistors are to be tested, one at a time in a random order to pe which of them are defective. Suppose that three of the five transistors are defective and let N denote the number of tests made until the first defective is spotted, and let No denote the number of additional tests until the second defective is Spotted Find the joint probability mass function of N1 and N2. 10. The joint probability density function of X and Y is given by S6,2) = (* +). xy f(x, y) = 0 < x < 1, 0 < y< 2 %3D (a) Verify that this is indeed a joint density function. (b) Compute the density function of X. (c) Find P{X > Y}. 11. Let X1, X2,. .., X, be independent random variables, each having a uniform dis- tribution over (0, 1). Let M = maximum (X1, X2,...,X). Show that the distri- bution function of M is given by FM(x) = x", 0*1 What is the probability density function of M?
Chapter 4 Random Variables and Expectation 137 6. The amount of time, in hours, that a computer functions before breaking down is a continuous random variable with probability density function given by de/100 f(x)%3D What is the probability that a computer will function between 50 and 150 hours before breaking down? What is the probability that it will function less than 100 hours? 7. The lifetime in hours of a certain kind of radio tube is a random variable having a probability density function given by x 100 f (x) = 100 *> 100 What is the probability that exactly 2 of 5 such tubes in a radio set will have to be replaced within the first 150 hours of operation? Assume that the events E, i= 1, 2, 3, 4, 5, that the ith such tube will have to be replaced within thi time are independent. 8. If the density function of X equals Ice-2 0 x<00 f(x) = { x <0 find c. What is P{X > 2}? 9. A set of five transistors are to be tested, one at a time in a random order to pe which of them are defective. Suppose that three of the five transistors are defective and let N denote the number of tests made until the first defective is spotted, and let No denote the number of additional tests until the second defective is Spotted Find the joint probability mass function of N1 and N2. 10. The joint probability density function of X and Y is given by S6,2) = (* +). xy f(x, y) = 0 < x < 1, 0 < y< 2 %3D (a) Verify that this is indeed a joint density function. (b) Compute the density function of X. (c) Find P{X > Y}. 11. Let X1, X2,. .., X, be independent random variables, each having a uniform dis- tribution over (0, 1). Let M = maximum (X1, X2,...,X). Show that the distri- bution function of M is given by FM(x) = x", 0*1 What is the probability density function of M?
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Transcribed Image Text:Chapter 4 Random Variables and Expectation
137
6. The amount of time, in hours, that a computer functions before breaking down
is a continuous random variable with probability density function given by
de/100
f(x)%3D
What is the probability that a computer will function between 50 and 150 hours
before breaking down? What is the probability that it will function less than
100 hours?
7. The lifetime in hours of a certain kind of radio tube is a random variable having
a probability density function given by
x 100
f (x) =
100
*> 100
What is the probability that exactly 2 of 5 such tubes in a radio set will have
to be replaced within the first 150 hours of operation? Assume that the events
E, i= 1, 2, 3, 4, 5, that the ith such tube will have to be replaced within thi
time are independent.
8. If the density function of X equals
Ice-2 0 x<00
f(x) = {
x <0
find c. What is P{X > 2}?
9. A set of five transistors are to be tested, one at a time in a random order to pe
which of them are defective. Suppose that three of the five transistors are defective
and let N denote the number of tests made until the first defective is spotted, and
let No denote the number of additional tests until the second defective is Spotted
Find the joint probability mass function of N1 and N2.
10. The joint probability density function of X and Y is given by
S6,2) = (* +).
xy
f(x, y) =
0 < x < 1, 0 < y< 2
%3D
(a) Verify that this is indeed a joint density function.
(b) Compute the density function of X.
(c) Find P{X > Y}.
11. Let X1, X2,. .., X, be independent random variables, each having a uniform dis-
tribution over (0, 1). Let M = maximum (X1, X2,...,X). Show that the distri-
bution function of M is given by
FM(x) = x", 0*1
What is the probability density function of M?
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