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- X and Y are continuous random variables with pdf f(x,y) = 2x for0 ≤x ≤y ≤1, and f(x,y) = 0 otherwise. Find the conditional expectation ofY given X = x.Let Mx, y be the moment generating function of random variables that are not independent of X and Y. Which of the following / which are not the properties of the function Mx, y?As soon as possible! Let X and Y be continuous random variables with joint PDF
- Let Y be a continuous random variable. Let c be a constant. PROVE Var (Y) = E (Y2) - E (Y)2If X, Y are normally distributed independent random variables, then show that W = 2X - Y is normally distributed. Do not use moment generating function, only use convolution formula.The distribution of a random vector is given by a function:f(x,y) = y/2 - x/2 ; for 0 < x < 1 and 2 < y < 3 = 0 ; otherwise Determine P( X > Y - 2), and sketch the graf.
- According to the Maxwell–Boltzmann law of theoret-ical physics, the probability density of V, the velocity of a gas molecule, isf(v) =⎧⎨⎩kv2e−βv2for v > 00 elsewhere where β depends on its mass and the absolute tem-perature and k is an appropriate constant. Show that the kinetic energy E = 1 2mV2, where m the massof the molecule is a random variable having a gammadistribution.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.LetXbea randomvariable havingthe uniformdistribution ontheinterval(θ,θ+1),θ∈R.Showthat Xis not complete.
- Prove that cov(X, Y) = cov(Y, X) for both discreteand continuous random variables X and Y.Suppose that the lifetime, X, and brightness, Y, of a light bulb are modeled as continuous random variables. Let their joint pdf be given by:f(x,y)=λ_1λ_2e^{-λ_1x-λ_2y},x,y>0 •Are lifetime and brightness independent?•Are lifetime and brightness uncorrelated?Find the moment generating function of the continuous random variable X∼U (a, b).Please give me the all details
