Let X and Y be independent integrable random variables on a probability space and f be a nonnegative convex function. Show that E[f(X +Y)] ≥ E[f(X +EY)]
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Let X and Y be independent integrable random variables on a probability space and f be a nonnegative convex
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- X and Y are independent, X ~ U (0, 1) and Y U (0, 1). Calculate the probability P(X+Y<1.5).Let X be a random variable on a closed and bounded interval [a, b] whose CDF is continuous. Let g(x) be a convex function. Prove that g (E (X)) ≤ E(g(X))Let X₁, X₂,..., X, be independent, uniformly distributed random variables on the interval [0, b]. a) Find the probability distribution function of X(n) = max(X₁, X₂,..., Xn). Fx (t) = for 0 ≤t≤ b and zero elsewhere I b) Find the probability density function of X(n)- fx (t) = c) Find the expected value of X(n) E(X(n)) = for 0 ≤t≤ b and zero elsewhere
- A) 0.4625 B) 0.3375 C) 0.600 D) 0.4355b) Suppose that X₁ and X₂ have the joint probability density function defined as f(x₁, x₂) = (x1x²,0x₁51, 0 ≤ x₂ ≤1 elsewhere Find: i) the value of w that makes f(x₁, x₂) a probability density function. ii) the joint cumulative distribution function for X₁ and X₂. iii) P (X₂ ≤ | X₂ ≤ ³).Let XE {-1,0, 1} . That is, X is a discrete random variable only takes three values -1, 0, and 1. Suppose the equality for Chebyshev's inequality holds for X and P(X = 0) = 0.3 , find P(X = 1) Let X be a random variable with probability density f(x) = x6 for x > 1 and O else. Use Chebyshev's inequality to bound P(X > 2.5) . Round your answer to 3 decimal places.
- Let X be a continuous random variable with cdf F(x). Show that E(I(X < x)) = F(r) where I is the indicator function (1 if XLet X be a random variable on a closed and bounded interval [a, b]. Let g(x) be a convex function. Prove that g(E(X)) ≤ E (g(X)Let X be a point randomly selected from the unit interval [0, 1]. Consider the random variable Y = (1-X)-¹/2 (a) Sketch Y as a function of X. (b) Find and plot the cdf of Y. (c) Derive the pdf of Y. (d) Compute the following probabilities: P(Y > 1), P(3 < Y < 6), P(Y ≤ 10).How do I prove this?8. Assume that X is a continuous random variable with pdf 1 if -6Let X be a continuous random variable with probability density function xe-X x20 fxlx) = 0 otherwise The value of a Select one: a. can be any positive real number b. must be 1 c. can be any real number d. can be any positive integerSEE 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