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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.X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2Let 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)
- Consider a random variable X with E[X] = 10, and X being positive. Estimate E[ln square root(X)] using Jensen’s inequality.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 continuous random variable with a pdf f(x) = { kx5 0≤x≤1, 0 elsewhere Determine the value of k.
- If we let RX(t) = ln MX(t), show that R X(0) = μ and RX(0) = σ2. Also, use these results to find the mean and the variance of a random variable X having the moment-generating function MX(t) = e4(et−1)If X has a standard normal distribution (X ∼ N(0, 1)), then the characteristic function of X is ϕX(t) = e −0.5(t^2) . Using this fact, show that X has skewness 0 and kurtosis 3Consider a series xt generated by the moving average process as: xt = µ + εt + θ1εt−1, where εt are independently identically distributed random variables with E(εt) = 0, and V ar(εt) = σ2. Calculate the unconditional mean and the unconditional variance of xt. What is meant by saying that a process like xt is invertible? What condition would assure that xt is invertible? If θ = 0.75, does xt satisfy the invertibility condition? What shapes of the ACF and PACF functions do you expect for xt? Derive the first 4 autocorrelations for this process (τ1 up to τ4). Carefully write the equations for the 1, 2, 3 and 4 step ahead forecasts for xt.
- The pdf of a continuous random variable is defined as f(x)= 3k/x3 over [10, ∞) and 0 elsewhere. Find the value of k that makes f(x) a valid pdf. (Round your answer to three decimal places)The pdf of a continuous random variable is defined as f(x)=8.75/x2 over (8.75, ∞) and 0 elsewhere. Find P( X > 65). (Round your answer to three decimal places).Let Y be a continuous random variable. Let c be a constant. PROVE Var (Y) = E (Y2) - E (Y)2
