4. The moment generating function (mgf) of a random variable X is given by mx(t) = (1 – t)-2, where t < 1. Determine the following: 4.1 E[X] 4.2 Var(x)
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A: From given data we have : λ=4 and Y~exp(λ)
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- Let X1, X2, ..., Xn be a random sample from a normal distribution with mean u and variance o?. Find an unbiased estimator for o? and show that ΣΧ-Χ- E(X; - X)² = E(X²) – nX². i=1 i=11. Let X be a Poisson random variable on the non-negative integers with rate λ = 4. Let W = 2X + 10. (a) What is the range of W? (b) Find a formula for Pw(k).(24) Let X be a random variable with p.d.f. Ze-2 07. a) Suppose that X is a uniform continuous random variable where 01. Which of the following is FALSE for a continuous random variable X with pdf y = f(x)? %3D A. P(6 S XS 9) = P(6 < X< 9) B. P(6 s XS 9) = f(6)+f(7)+f(8)+f(9) %3D C. P(6 < XS9) = area under the curve y=f(x) between 6 and 9 D. P(X = 9) = 0Let a random variable X has the following function:FX(x) = { 0 ; x < 0(1/2)+(x/2) ; 0 ≤ x < 11 ; x ≥ 1. Verify that FX(x) is a CDF. Determine (i) the PDF of X, (ii) E[e^X], (iii) P[X = 0|X≤ 0.5].Let X be a random variable taking positive values and assume that E(X) exists. Which of the following statements are true? • (i) E(X*) > (E(X))*. • (ii) E(1/X²) > 1/E(X²). • (ii) E(e-3X) > e°der a random sample X1, X2, ..., X, having the pdf, f(r; 0) 1X is a random variable that has the following PDF function: fx(x) = 0.28(x + 2) + 0.38(x + 1) + 0.358(x) + 0.158(x – 1) %3D For y = x2, find: 1- fy) 2- Fy(y) 3- E[y] 4- oX is an exponential random variable with parameter λ = 1/4 Calculate P {X ≤ 4}4. Let X be a positive random variable (i.e. P(X 1/E(X) (b) E(-log(X)) > -log(E(X)) (c) E(log(1/X))> log(1/E(X)) (d) E(X³) > (E(X))³8. Let the random variable X have the pdf 2 x2 fx (x) = exp %3D - V2n 2 Find the mean and the variance of X. Hint: Compute E (X) directly and E (X²) by comparing the integral with the integral representing the variance of a random variable that is N(0,1). i DCO 04 < (X - 5)2 < 38.4).SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. FreemanMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. Freeman