Concept explainers
Find the expected values and the standard deviations (by inspection) of the normal random variables with the density functions given in Exercises 15–18.
Want to see the full answer?
Check out a sample textbook solutionChapter 12 Solutions
Calculus & Its Applications
- Identify the distributions of the random variables with the moment- generating functions shown below. For each random variable also indicate what the mean and the variance are. a) m(t) = e22(e'–1) b) m(t) = 1/(1 – 2t)² %3D с) т() — 4t d) m(t) = (0.3e + 0.7)18arrow_forwardExercise 3. Let X be a random variable with mean µ and variance o². For a € R, consider the expectation E((X − a)²). a) Write E((X - a)²) in terms of a, μ and σ². b) For which value a is E((X − a)²) minimal? c) For the value a from part (b), what is E((X − a)²)?arrow_forwardLet X ∼ U(a, b). Use the definition of the variance of a continuous random variable (Equation 2.38 or 2.39) to show that . σ²X = (b-a)²/12.arrow_forward
- 1)Let x be a random variable Gaussian with zero mean and variance 1. Find:a)The conditional pdf and pdf of x given x > 0;b)E [ x| x>0 ]c)Var [ x | x >0]arrow_forward1.4 Let X be a continuous random variable with pdf, fx(x), and fx(t+5) = fx(5 – t) for all t> 0. Please provide the mean of X.arrow_forwardPart C and Darrow_forward
- 3. Let X be an exponentially distributed random variable with parameter 2. a) Find and sketch the cdf and pdf of Y = 2- 3X. b) Find the mean and variance of Y. c) Find the distribution of Z = VX. Calculate its mean and variance in Mathematica. d) Obtain characteristic functions of X and Y.arrow_forwardPart 1: Prove that the expectation and the variance of a lognormal random variable X with mean u, standard deviation o and a normal random variable W with mean 0, standard deviation o has the following relation: (Note: You must prove the relation from both sides!) E(X)= µ=e®+*2 O = Jln +1 Var (X)=o = e0+* (e -1) 0 = In u-arrow_forward1. The incidence of cancer among phosphate mine workers and their exposure to radiation was investigated. Cancer incidence was recorded among 60 phosphate mine workers who had 0-29 working level months (WLM) of exposure, 60 phosphate mine workers who had 30-89 WLM of exposure, and 60 phosphate mine workers who had 90-120 WLM of exposure. The number of cancer incidences for each group are presented in Table 1. Table 1: Two-way Table of Counts Cancer Incidence Exposure Level Yes No Total 0-29 WLM 20 40 60 30-89 WLM 33 27 60 90-120 WLM 43 17 60 Total 96 84 180 Perform a x? test, at the 5% level of significance, whether cancer incidence rates differ between the three levels of radiation among phosphate mine workers.arrow_forward
- B) Let the random variable X have the moment generating function e3t M(t) for -1arrow_forwardThe pH water samples from a specific lake is a random variable X with probability density function. f(x) = {8 (7– x)², 5 < x<7 0, elsewhere a. ) Find E(X) ( expected value or mean) b. ) Find V(X) ( Variance)arrow_forwardb) Consider a random variable X which takes on values 1 and 2 with probability 0.25 and 0.75, respectively (i.e. P[x = 1] = 0.25 and P[x = 2] = 0.75 i) Determine the mean and variance of X. ii) Plot the probability density function (pdf) and probability distribution function (PDF) of Xarrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning