Find E[X], and Var[X], when "X" is a normal random variable with pa- rameter µ and o? respectively.
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A: Hello! As you have posted more than 3 sub parts, we are answering the first 3 sub-parts. In case…
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- Consider a random variable X with E[X] = 10, and X being positive. Estimate E[ln√X] using Jensen’s inequality.For an exponential random variable (X) having θ = 4 and pdf given by: f(x) = (1/θ)e^(−x/θ ) where x ≥ 0, compute the following: a) E(X). b) Var(X). c) P(X > 3).Consider a random variable X with E[X] = 10, and X being positive. Estimate E[ln square root(X)] using Jensen’s inequality.
- For any continuous random variables X, Y , Z and any constants a, b, show the following from the definition of the covariance:Consider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?For a random variable (X) having pdf given by: f(x) = (k)x^3 where 0 ≤ x ≤ 1, compute the following: a) k b) E(X). c) Var(X). d) P(X > 0.25).
- Let X be an exponential random variable with standard deviation σ. FindP(|X − E(X)| > kσ ) for k = 2, 3, 4, and compare the results to the boundsfrom Chebyshev’s inequality.Prove that if M(t) is the MGF of a random variable X, then the MGF of a + bX is e^at M (bt)Use the moment generating function technique to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Exponential(θ), for i =1, . . . , n. Find the distribution of Y = X1 + · · · + Xn.
- Let X be a random variable with the mean µ and the variance. Show that E|X|=3 and E|X2|=13, use the Chebyshev inequality to determine a lower bound for P[-2 < X < 8].If the probability density of X is given by f(x) =kx3(1 + 2x)6 for x > 00 elsewhere where k is an appropriate constant, find the probabilitydensity of the random variable Y = 2X 1 + 2X . Identify thedistribution of Y, and thus determine the value of k.Let X be a random variable with pdff(x) = 4x^3 if 0 < x < 1 and zero otherwise. Use thecumulative (CDF) technique to determine the pdf of each of the following random variables: 1) Y=X^4, 2) W=e^(-x) 3) Z=1-e^(-x) 4) U=X(1-X)