Concept explainers
The lengths of life Y for a type of fuse has an exponential distribution with a density
- a If two such fuses have independent life lengths Y1 and Y2, find their joint probability density function.
- b One fuse from part (a) is in a primary system, and the other is in a backup system that comes into use only if the primary system fails. The total effective life length of the two fuses, therefore, is Y1 + Y2. Find P(Y1 + Y2 ≤ a), where a > 0.
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Chapter 5 Solutions
Mathematical Statistics with Applications
- A Troublesome Snowball One winter afternoon, unbeknownst to his mom, a child bring a snowball into the house, lays it on the floor, and then goes to watch T.V. Let W=W(t) be the volume of dirty water that has soaked into the carpet t minutes after the snowball was deposited on the floor. Explain in practical terms what the limiting value of W represents, and tell what has happened physically when this limiting value is reached.arrow_forwardIn 1950 an experiment was done observing the time gaps between successive cars on the Arroyo Seco Freeway. The data show that the density function of these time gaps was given approximately by p(t)=ae−0.119t where t is the time in seconds and a is a constant.Find the constant a.arrow_forward. The thickness of a wooden shelf in mm, has probability density function f(x) = {0.75 – 0.75 (x – 5)2} : for 4</= x </ =6 and 0 otherwise Show the table of the points for x and y that are to be plotted on the graph of the function above.arrow_forward
- Suppose the variables Q and W are skewed distributions defined over a limited set of values. What is the probability that W takes on a value between 0.5 and 1.75? What is E(W)? Suppose that the domain of W changes to 0<t<2.5, what happens to the Probability Density Function (PDF)?arrow_forwardA college professor never finishes his lecture before the end of the hour and always finishes his lectures within 2 min after the hour. Let X = the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of X is f(x)= {kx2O < x < 2} 0 otherwise a. Find the value of k and draw the corresponding density curve.[ Hint: Total area under the graph of f (x) is 1.] b. What is the probability that the lecture ends within 1min of the end of the hour? c. What is the probability that the lecture continues beyond the hour for between 60 and 90 sec? d. What is the probability that the lecture continues for at least 90 sec beyond the end of the hour?arrow_forwarda. Find the 50-th percentile of X. That is to say the value of x such that P (X ≤ x) = 0.5. b. Now say you have two independent jet engines. What is the probability that only one of themwill last more than 12 months before needing to be rebuilt? c. Find the probability density function f(x) by taking the derivative of F(x) with respect to x.arrow_forward
- A continuous random variable X that can assume values between x = 2 and x = 5 has a density function given by f(x) =2(1 + x)/27. Find: (a) P(X < 4); (b) P(3 ≤ X < 4).arrow_forwardThe joint probability density function of X and Y is constant over the shaded region in the attached graph, and otherwise = 0. a) Find the value of f (x, y) for (x, y) inside the shaded region. b) Find the probability that both X and Y are between 0 and 1.arrow_forwardFor a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)arrow_forward
- let x and y have joint density function f(x,y) = 2e ^-x-y 0<x<y<∞ are they independent? find the marginal density function,their covariance and correlation coefficientarrow_forwardX, Y follow the exponential distribution with rate parameter lambda = 5.0 P[X >= 5] ? Write P[X+Y <= 14] as an integralarrow_forwardA random variable X has probability density function (pdf) fx(x), wherefx(x) =and k is a constant. Find:i. kii. P [ X > 1]iii. the mean of Xarrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning