Problem 5.1P: Let X be a random variable with probability density function f(x)={c(1x2)1x00otherwise a. What is... Problem 5.2P Problem 5.3P Problem 5.4P: The probability density function of X. the lifetime of a certain type of electronic device (measured... Problem 5.5P Problem 5.6P: Compute E[X] if X has a density function given by a. f(x)={14xex2x00otherwise b.... Problem 5.7P: The density function of X is given by f(x)={a+bx20x10otherwise . If E[X]=35, find a and b. Problem 5.8P: The lifetime in hours of an electronic tube is a random variable having a probability density... Problem 5.9P: Consider Example 4b &I of Chapter 4 &I, but now suppose that the seasonal demand is a continuous... Problem 5.10P: Trains headed for destination A arrive at the train station at 15-minute intervals starting at 7... Problem 5.11P: A point is chosen at random on a line segment of length L. Interpret this statement, and find the... Problem 5.12P: A bus travels between the two cities A and B. which are 100 miles apart. If the bus has a breakdown,... Problem 5.13P: You arrive at a bus stop at 10A.M., knowing that the bus will arrive at some time uniformly... Problem 5.14P: Let X be a uniform (0, 1) random variable. Compute E[Xn] by using Proposition 2.1, and then check... Problem 5.15P: If X is a normal random variable with parameters =10 and 2=36, compute a. P{X5}; b. P{4X16}; c.... Problem 5.16P: The annual rainfall (in inches) in a certain region is normally distributed with =40 and =4. What is... Problem 5.17P: The salaries of physicians in a certain speciality are approximately normally distributed. If 25... Problem 5.18P: Suppose that X is a normal random variable with mean 5. If P{X9}=.2, approximately what is Var(X)? Problem 5.19P: Let be a normal random variable with mean 12 and variance 4. Find the value of c such that... Problem 5.20P: If 65 percent of the population of a large community is in favor of a proposed rise in school taxes,... Problem 5.21P: Suppose that the height, in inches, of a 25-year-old man is a normal random variable with parameters... Problem 5.22P: Every day Jo practices her tennis serve by continually serving until she has had a total of 50... Problem 5.23P: One thousand independent rolls of a fair die will be made. Compute an approximation to the... Problem 5.24P: The lifetimes of interactive computer chips produced by a certain semiconductor manufacturer are... Problem 5.25P: Each item produced by a certain manufacturer is, independently, of acceptable quality with... Problem 5.26P: Two types of coins are produced at a factory: a fair coin and a biased one that comes up heads 55... Problem 5.27P: In 10,000 independent tosses of a coin, the coin landed on heads 5800 times. Is it reasonable to... Problem 5.28P: Twelve percent of the population is left handed. Approximate the probability that there are at least... Problem 5.29P: A model for the movement of a stock supposes that if the present price of the stock is s, then after... Problem 5.30P: An image is partitioned into two regions, one white and the other black. A reading taken from a... Problem 5.31P: a. A fire station is to be located along a road of length A,A. If fires occur at points uniformly... Problem 5.32P: The time (in hours) required to repair a machine is an exponentially distributed random variable... Problem 5.33P: If U is uniformly distributed on (0,1), find the distribution of Y=log(U). Problem 5.34P: Jones figures that the total number of thousands of miles that a racing auto can be driven before it... Problem 5.35P Problem 5.36P: The lung cancer hazard rate (t) of a t-year-old male smoker is such that (t)=.027+.00025(t40)2t40... Problem 5.37P: Suppose that the life distribution of an item has the hazard rate function (t)=t3,t0. What is the... Problem 5.38P: If X is uniformly distributed over (1,1), find (a) P{|X|12} (b) the density function of the random... Problem 5.39P Problem 5.40P: If X is an exponential random variable with parameter =1, compute the probability density function... Problem 5.41P: If X is uniformly distributed over(a,b), find a and b if E[X]=10,var(X)=48. Problem 5.42P Problem 5.43P: Find the distribution of R=Asin, where A is a fixed constant and is uniformly distributed on (2,2).... Problem 5.44P: Let Y be a log normal random variable (see Example 7e for its definition) and let c0 be a constant.... Problem 5.1TE: The speed of a molecule in a uniform gas at equilibrium is a random variable whose probability... Problem 5.2TE: Show that E[Y]=0P{Yy}dy0P{Yy}dy Hint: Show that 0P{Yy}dy=0xfY(x)dx0P{Yy}dy=0xfY(x)dx Problem 5.3TE: Show that if X has density function f. then E[g(X)]=g(x)f(x)dx Hint: Using Theoretical Exercise 5.2,... Problem 5.4TE Problem 5.5TE: Use the result that for a nonnegative random variable E[Y]=0P{Yt}dt to show that for a nonnegative... Problem 5.6TE Problem 5.7TE: The standard deviation of X. denoted SD(X), is given by SD(X)=Var(X). Find SD(aX+b) if X has... Problem 5.8TE: Let X be a random variable that takes on values between 0 and c. That is,p{0Xc}=1 .Show that... Problem 5.9TE: Show that Z is a standard normal random variable; then, for x0. a. P{Zx}=P{Zx} b. P{|Z|x}=2P{Zx} c.... Problem 5.10TE: Let f(x) denote the probability density function of a normal random variable with mean , and... Problem 5.11TE: Let Z be a standard normal random variable Z and let g be a differentiable function with derivative... Problem 5.12TE: Use the identity of Theoretical Exercises 5.5 . Problem 5.13TE: The median of a continuous random variable having distribution function F is that value m such that... Problem 5.14TE: The mode of a continuous random variable having density f is the value of x for which f (x) attains... Problem 5.15TE: If X is an exponential random variable with parameter , and c0, show that cX is exponential with... Problem 5.16TE: Compute the hazard rate function of X when X is uniformly distributed over (0, a). Problem 5.17TE: If X has hazard rate function X(t), compute the hazard rate function of aX where a is a positive... Problem 5.18TE Problem 5.19TE: If X is an exponential random variable with mean 1, show that E[Xk]=k!kk=1,2,... Hint: Make use of... Problem 5.20TE Problem 5.21TE Problem 5.22TE: Compute the hazard rate function of a gamma random variable with parameters (,) and show it is... Problem 5.23TE: Compute the hazard rate function of a Weibull random variable and show it is increasing when 1 and... Problem 5.24TE Problem 5.25TE: Let Y=(Xv) Show that if X is a Weibull random variable with parameters v,, and , then Y is an... Problem 5.26TE: Let F be a continuous distribution function. If U is uniformly distributed on (0,1), find the... Problem 5.27TE: If X is uniformly distributed over (a,b), what random variable, having a linear relation with X. is... Problem 5.28TE: Consider the beta distribution with parameters (a,b). Show that a. when a1 and b1, the density is... Problem 5.29TE Problem 5.30TE Problem 5.31TE Problem 5.32TE: Let X and Y be independent random variables that are both equally likely to be either 1,2,...,(10)N... Problem 5.33TE Problem 5.1STPE: The number of minutes of playing time of a certain high school basketball player in a randomly... Problem 5.2STPE: For some constant c. the random variable X has the probability density function... Problem 5.3STPE Problem 5.4STPE Problem 5.5STPE: The random variable X is said to be a discrete uniform random variable on the integers 1.2..... n if... Problem 5.6STPE Problem 5.7STPE: To be a winner in a certain game, you must be successful in three successive rounds. The game... Problem 5.8STPE: A randomly chosen IQ test taker obtains a score that is approximately a normal random variable with... Problem 5.9STPE: Suppose that the travel time from your home to your office is normally distributed with mean 40... Problem 5.10STPE: The life of a certain type of automobile tire is normally distributed with mean 34,000 miles and... Problem 5.11STPE: The annual rainfall in Cleveland, Ohio, is approximately a normal random variable with mean 40.2... Problem 5.12STPE Problem 5.13STPE Problem 5.14STPE Problem 5.15STPE: The number of years that a washing machine functions is a random variable whose hazard rate function... Problem 5.16STPE Problem 5.17STPE Problem 5.18STPE: There are two types of batteries in a bin. When in use, type i batteries last (in hours) an... Problem 5.19STPE Problem 5.20STPE: For any real number y define byy+=y,ify00,ify0 Let c be a constant. a. Show that... Problem 5.21STPE: With (x) being the probability that a normal random variable with mean 0 and variance 1 is less than... Problem 5.22STPE Problem 5.23STPE: Letf(x)={13ex1313e(x1)ifx0if0x1ifx1 a. Show that f is a probability density function. (That is, show... Problem 5.24STPE format_list_bulleted
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON