Theorem 5-6. (c) Distribution of the Product of two Random Variables. Let X and Y be two independent continuous r.v.'s. Then the p.d.f. of their product U = XY, is given by : h (u) = [ fa(x, ") dx (5-31) | fxr x, ... %3D
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Prove the theorem
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- Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0Use the moment generating function to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Poiss(λi), for i = 1, . . . , n.Find the distribution of Y = X1 + · · · + Xn.LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.
- If a random variable X has a discrete uniform distribution. fx(x)=1/k for x=1,2,..,k;0 otherwise. Derive P.G.F of X and compute E(2x+1)9.19 Let X and Y be two continuous random variables, with joint proba- bility density function f(x, y): - 30 -50x²-50y² +80xy for -(b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If yes, give a proof; if no, give a counterexample.
- For any continuous random variables X, Y , Z and any constants a, b, show the following from the definition of the covariance:If X1, X2, ... , Xn constitute a random sample of size nfrom a geometric population, show that Y = X1 + X2 +···+ Xn is a sufficient estimator of the parameter θ.The distribution of the random variable X is determined by the function: f(x) = ((x-1)2)/c ; for 1<x<3 = 0 ; inak Sketch the graph of f(x) Determine the FY(y) and fY(y) for Y = X^2 - 1