Concept explainers
Emergency Room Traffic. Desert Samaritan Hospital in Mesa, Arizona, keeps records of its emergency-room traffic. Beginning at 6:00 P.M. on any given day, the elapsed time, in hours, until the first patient arrives is a variable with density curve y = 6.9e−6.9x for x > 0, and y = 0 otherwise. Here e is Euler’s number, which is approximately 2.71828. Most calculators have an e-key. Using calculus, it can be shown that the area under this density curve to the left of any number x greater than 0 equals 1 − e−6.9x.
- a. Graph the density curve of this variable.
- b. What percentage of the time does the first patient arrive between 6:15 P.M. and 6:30 P.M.?
Want to see the full answer?
Check out a sample textbook solutionChapter 6 Solutions
Introductory Statistics (10th Edition)
Additional Math Textbook Solutions
Statistics for Psychology
Elementary Statistics ( 3rd International Edition ) Isbn:9781260092561
Probability and Statistics for Engineering and the Sciences
Elementary Statistics Using Excel (6th Edition)
Basic Business Statistics, Student Value Edition (13th Edition)
- 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_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_forwardGiven the density function f(x) = 2(1 − x), for 0 < x < 1, = 0, otherwise. Find the standard deviationarrow_forward
- The density function is often used as a model for the lengths of life of physical systems. Suppose Y has the Weibull density just given. Find: a) the density function of U = Ym b) E(Yk) for any positive integer karrow_forwardThe lifetime, X, of a particular integrated circuit has an exponential distribution with rate of ?=0.5 per year. Thus, the density of X is:f(x,?) = ? e−?x for 0 ≤ x ≤ ∞, ? = 0.5 . ? is what R calls rate. Hint: This is a problem involving the exponential distribution. Knowing the parameter ? for the distribution allows you to easily answer parts a ,b ,c and use the built-in R functions for the exponential distribution (dexp(), pexp(), qexp()) for other parts . Or (not recommended) you should be able to use the R integrate command with f(x) defined as above or with dexp() for all parts.a) What is the expected value of X? b) What is the variance of X? c) What is the standard deviation of X? d) What is the probability that X is greater than its expected value? e) What is the probability that X is > 5? f) What is the probability that X is > 10? g) What is the probability that X > 10 given that X > 5? h) What is the median of X?arrow_forwardLet X denote 0.025 × the ambient air temperature (˚C) and let Y denote the time (min) that it takes for a diesel engine to warm up. Assume that (X, Y) has joint probability density function f(x,y) = 1.6x (1 − x)(6 + 5x − 4y), for 0 < x < 1, 0 < y < 0.5. While you cannot guess the value of the correlation from the regression curve for X or Y, do they suggest whether it likely is positive or negative?arrow_forward
- Example 2: For each part below, find the endpoints on a standard normal density (curves have X~N(0,1) ). The area to the left of the endpoint is about 0.20. The area between ± z is 0.80. The area to the right of the endpoint is 0.4arrow_forwardLet X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is the following. F(x) = 0 x < 0 x2 25 0 ≤ x < 5 1 5 ≤ x Use the cdf to obtain the following. (If necessary, round your answer to four decimal places.) (a) Calculate P(X ≤ 3). (b) Calculate P(2.5 ≤ X ≤ 3). (c) Calculate P(X > 3.5). (d) What is the median checkout duration ? [solve 0.5 = F()]. (e) Obtain the density function f(x). f(x) = F ′(x) = 15x 0 ≤ x < 5 0 otherwise (f) Calculate E(X). (g) Calculate V(X) and ?x. V(X) ?x (h) If the borrower is charged an amount h(X) = X2 when checkout duration is X, compute the expected charge E[h(X)].arrow_forwardA uniform density function for X over an interval of unit length is such that P(1/4<X<12/)=1/4. What is the left-hand endpoint of that interval of unit length? A 0 B 1/8 C 1/4 D 3/8 E Cannot be determined from the given informationarrow_forward
- Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is the following. F(x) = 0 x < 0 x2 16 0 ≤ x < 4 1 4 ≤ x Use the cdf to obtain the following. (If necessary, round your answer to four decimal places.) (a) Calculate P(X ≤ 3). (b) Calculate P(2.5 ≤ X ≤ 3). (c) Calculate P(X > 3.5). (d) What is the median checkout duration ? [solve 0.5 = F()]. (e) Obtain the density function f(x). f(x) = F ′(x) = 0 ≤ x < 4 0 otherwise (f) Calculate E(X). (g) Calculate V(X) and ?x. V(X)?x (h) If the borrower is charged an amount h(X) = X2 when checkout duration is X, compute the expected charge E[h(X)].arrow_forwardFind the mean, standard deviation and the median of the density function: f(x)= (6/343)x(7-x) [0,7]arrow_forwardThe distance, X, between consecutive anomalies on long cable has an exponential distribution with mean 12 meters. Thus, the density of X is:: f(x,?) = ? e−?x for 0 ≤ x ≤ ∞, ? = 1 12 . ? is what R calls rate.Hint: This is a problem involving the exponential distribution. Knowing the parameter ? for the distribution allows you to easily answer parts a ,b ,c and use the built-in R functions for the exponential distribution (dexp(), pexp(), qexp()) for other parts . Or (not recommended) you should be able to use the R integrate command with f(x) defined as above or with dexp() for all parts. ) What is the probability that X is larger than its expected value? e) What is the probability that X is > 13? f) What is the probability that X is > 14? g) What is the probability that X > 14 given that X > 13? h) What is the median of X?arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage LearningCollege AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning