Show that the characteristic function of a random variable X satisfies |1 - 0(t)| ≤ EtX.
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.Assume that the probability that an airplane engine will fail during a torture test is 12and that the aircraft in question has 4 engines. Construct a sample space for the torture test. Use S for survive and F for fail.Prove that for a continuous random variable X,E (aX+ b) = aE (X) + b.
- Suppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on interval [0, 1]. Determine the value of E[(X1-2X2+X3)2]Q1 Let X be a non-negative discrete random variable. Prove that the expected value of X satisfies E(X) = √ ** (1 – F(x)) da -Suppose X is a random variable, whose pdf is defined as follows: 2x = (²x) (u(x) - u(x − 3)) where u(x) is the unit step function. Determine the conditional pdf fx(x 1Let the random variable X have the moment generating function M(t) = e³t 1-t² = -1 < t < 1 What are the mean and the variance of X, respectively?B) Let the random variable X have the moment generating function e3t M(t) for -1Let u(X) be a nonnegative function of a random variable X. It can be shown that P(u(X) >= c) <= E(u(X)) / c Suppose X is a random variable with momonet generating function Mx(t). Show that P(X>= a) <= exp(-ta) * Mx(t) Thank you2. Let X and Y be two arbitrary random variables. Define the conditional variance of X given Y by Var[X|Y] = E[X²|Y] – (E[X|Y])². Show that Var[X|Y] ≥ 0 and the variance decomposition formula Var[X] = E[Var[X|Y]] + Var[E[X|Y]]Let X and Y be two discrete random variables and f(X) and f(Y) be some functions of X and Y respectively. Derive the most general inequality relationship between I(X;Y) and I(f(X); g(Y)).Let pX(x) be the pmf of a random variable X. Find the cdf F(x) of X and sketch its graph along with that of pX(x) if pX(x)=1, x=0, zero elsewhereSEE MORE QUESTIONSRecommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageCollege Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage LearningBig Ideas Math A Bridge To Success Algebra 1: Stu…AlgebraISBN:9781680331141Author:HOUGHTON MIFFLIN HARCOURTPublisher:Houghton Mifflin HarcourtGlencoe Algebra 1, Student Edition, 9780079039897…AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageCollege Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage LearningBig Ideas Math A Bridge To Success Algebra 1: Stu…AlgebraISBN:9781680331141Author:HOUGHTON MIFFLIN HARCOURTPublisher:Houghton Mifflin HarcourtGlencoe Algebra 1, Student Edition, 9780079039897…AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning