A First Course in Probability
9th Edition
ISBN: 9780321794772
Author: Sheldon Ross
Publisher: PEARSON
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Textbook Question
Chapter 5, Problem 5.35P
The lung cancer hazard rate
Assuming that a 40-year-old male smoker survives all other hazards, what is the
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The lung cancer hazard rate λ(t) of a t-year-old male smoker is such that λ(t) = 0.027 + 0.00025(t-40)2, and t >= 40. Assuming that a 40-year-old male smoker survives all other hazards,
(a) what is the probability that he survives to age 50 without contracting lung cancer?(b) what is the probability that he survives to age 60 without contracting lung cancer?
Suppose that the life distribution of an item has the hazard rate function 1(t) = t³,t > 0. What is the
probability that
a. the item survives to age 2?
b. the item's lifetime is between 0.4 and 1.4?
C.
a 1-year-old item will survive to age 2?
The failure time (T) of an electronic circuit board follows exponentially distribution
with failure rate λ = 10−4/h. What is the probability that
(i) it will fail before 1000 h?
(ii) it will survive at least 10,000 h ?
(iii)Determine the mean time to failure.
Chapter 5 Solutions
A First Course in Probability
Ch. 5 - Let X be a random variable with probability...Ch. 5 - Prob. 5.2PCh. 5 - Prob. 5.3PCh. 5 - The probability density function of X. the...Ch. 5 - Prob. 5.5PCh. 5 - Compute E[X] if X has a density function given by...Ch. 5 - The density function of X is given by...Ch. 5 - The lifetime in hours of an electronic tube is a...Ch. 5 - Consider Example 4b &I of Chapter 4 &I, but now...Ch. 5 - Trains headed for destination A arrive at the...
Ch. 5 - A point is chosen at random on a line segment of...Ch. 5 - A bus travels between the two cities A and B....Ch. 5 - You arrive at a bus stop at 10A.M., knowing that...Ch. 5 - Let X be a uniform (0, 1) random variable. Compute...Ch. 5 - If X is a normal random variable with parameters...Ch. 5 - The annual rainfall (in inches) in a certain...Ch. 5 - The salaries of physicians in a certain speciality...Ch. 5 - Suppose that X is a normal random variable with...Ch. 5 - Let be a normal random variable with mean 12 and...Ch. 5 - If 65 percent of the population of a large...Ch. 5 - Suppose that the height, in inches, of a...Ch. 5 - Every day Jo practices her tennis serve by...Ch. 5 - One thousand independent rolls of a fair die will...Ch. 5 - The lifetimes of interactive computer chips...Ch. 5 - Each item produced by a certain manufacturer is,...Ch. 5 - Two types of coins are produced at a factory: a...Ch. 5 - In 10,000 independent tosses of a coin, the coin...Ch. 5 - Twelve percent of the population is left handed....Ch. 5 - A model for the movement of a stock supposes that...Ch. 5 - An image is partitioned into two regions, one...Ch. 5 - a. A fire station is to be located along a road of...Ch. 5 - The time (in hours) required to repair a machine...Ch. 5 - If U is uniformly distributed on (0,1), find the...Ch. 5 - Jones figures that the total number of thousands...Ch. 5 - The lung cancer hazard rate (t) of a t-year-old...Ch. 5 - Suppose that the life distribution of an item has...Ch. 5 - If X is uniformly distributed over (1,1), find (a)...Ch. 5 - Prob. 5.38PCh. 5 - If X is an exponential random variable with...Ch. 5 - Prob. 5.40PCh. 5 - Find the distribution of R=Asin, where A is a...Ch. 5 - Let Y be a log normal random variable (see Example...Ch. 5 - The speed of a molecule in a uniform gas at...Ch. 5 - Show that E[Y]=0P{Yy}dy0P{Yy}dy Hint: Show that...Ch. 5 - Show that if X has density function f. then...Ch. 5 - Prob. 5.4TECh. 5 - Use the result that for a nonnegative random...Ch. 5 - Prob. 5.6TECh. 5 - The standard deviation of X. denoted SD(X), is...Ch. 5 - Let X be a random variable that takes on values...Ch. 5 - Show that Z is a standard normal random variable;...Ch. 5 - Let f(x) denote the probability density function...Ch. 5 - Let Z be a standard normal random variable Z and...Ch. 5 - Use the identity of Theoretical Exercises 5.5 .Ch. 5 - The median of a continuous random variable having...Ch. 5 - The mode of a continuous random variable having...Ch. 5 - If X is an exponential random variable with...Ch. 5 - Compute the hazard rate function of X when X is...Ch. 5 - If X has hazard rate function X(t), compute the...Ch. 5 - Prob. 5.18TECh. 5 - If X is an exponential random variable with mean...Ch. 5 - Prob. 5.20TECh. 5 - Prob. 5.21TECh. 5 - Compute the hazard rate function of a gamma random...Ch. 5 - Compute the hazard rate function of a Weibull...Ch. 5 - Prob. 5.24TECh. 5 - Let Y=(Xv) Show that if X is a Weibull random...Ch. 5 - Let F be a continuous distribution function. If U...Ch. 5 - If X is uniformly distributed over (a,b), what...Ch. 5 - Consider the beta distribution with parameters...Ch. 5 - Prob. 5.29TECh. 5 - Prob. 5.30TECh. 5 - Prob. 5.31TECh. 5 - Let X and Y be independent random variables that...Ch. 5 - Prob. 5.33TECh. 5 - The number of minutes of playing time of a certain...Ch. 5 - For some constant c. the random variable X has the...Ch. 5 - Prob. 5.3STPECh. 5 - Prob. 5.4STPECh. 5 - The random variable X is said to be a discrete...Ch. 5 - Prob. 5.6STPECh. 5 - To be a winner in a certain game, you must be...Ch. 5 - A randomly chosen IQ test taker obtains a score...Ch. 5 - Suppose that the travel time from your home to...Ch. 5 - The life of a certain type of automobile tire is...Ch. 5 - The annual rainfall in Cleveland, Ohio, is...Ch. 5 - Prob. 5.12STPECh. 5 - Prob. 5.13STPECh. 5 - Prob. 5.14STPECh. 5 - The number of years that a washing machine...Ch. 5 - Prob. 5.16STPECh. 5 - Prob. 5.17STPECh. 5 - There are two types of batteries in a bin. When in...Ch. 5 - Prob. 5.19STPECh. 5 - For any real number y define byy+=y,ify00,ify0 Let...Ch. 5 - With (x) being the probability that a normal...Ch. 5 - Prob. 5.22STPE
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- Suppose that the distribution of the lifespan of an item has the hazard rate function λ(t) = t3. What is the probability that: (a) the item survives to age 2? (b) a 1-year-old item will survive to age 2?arrow_forwardSuppose that an item has a hazard rate function X(t) = 2.4t², t> 0. What is the probability that (a) the item survives to age 0.5? (b) the item's lifetime is between 1.5 and 3? (c) a (-year-old item will survive to age 3?arrow_forwardThe failure time (T) of an electronic circuit board follows exponentially distribution with failure rate λ = 10−4/h. What is the probability that (i) it will fail before 1000 h? (ii) it will survive at least 10,000 h ? (iii) it will fail between 1000 and 10,000 h? (iv)Determine the mean time to failure. .arrow_forward
- The pdf of the time to failure of an electronic component in a copier (in hours) is f (x) = [exp (-x/3100)]/3100 for x > 0 and f (x) = 0 for x ≤ 0. Determine the probability that:(a) A component lasts more than 1180 hours before failure.(b) A component fails in the interval from 1180 to 2010 hours.(c) A component fails before 3010 hours.(d) Determine the number of hours at which 11% of all components have failed.(e) Determine the mean.Round your answers to three decimal places (e.g. 98.765).arrow_forwardThe probability that a vehicle will enter an intersection within a period of x minutes is given by P(x) = 1 -e-0.1x Determine symbolically the elapsed time x when there is a 50-50 chance that the vehicle has entered the intersection. The elapsed time is approximately min. (Type an integer or decimal rounded to two decimal places as needed.)arrow_forwardNext question When a man observed a sobriety checkpoint conducted by a police department, he saw 662 drivers were screened and 9 were arrested for driving while intoxicated. Based on those results, we can estimate that P(W) = 0.01360, where W denotes the event of screening a driver and getting someone who is intoxicated. What does P (W) denote, and what is its value? What does P (W) represent?arrow_forward
- The number of work accidents that occur per week in a mine follows Poisson's law so that the probability of 2 accidents occurring is equal to 3/2 of the probability of an accident occurring. Calculate the probability that the time for the first accident to occur is one week.arrow_forwardA individual, with a utility function lnW and an initial wealth level W0, is faced with a fair gamble of winning or losing $h (where W0 > h > 0) with 50-50 chance. (a) Is this individual risk averse? Explain.(b) Suppose that the individual is willing to pay up to an amount of f in order to avoid such a gamble. Give the equation that determines f, and solve the equation for f (i.e., express f in terms of W0 and h). (c) Show that f increases as h increases.arrow_forwardThe lifetime, t, in hours, of an alkaline battery is exponentially distributed with ? = 0.10 per hour. What is the probability that the battery will last more than 18 hours?arrow_forward
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