Suppose that X~X(v) and Y~X"(n). If μx = 6,0 y a) State with parameter(s) the distribution of W. b) Find the mean and variance of W. c) Find the P(W < 4.65). d) Compute the P(W > 0.2411). e) Compute the P(0.1786 < W<3.581). 16 and then
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- Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. Find (a) fXY(x, y), (b) fYZ(y, z), (c) fZ(z)Consider a random variable Y with PDF Pr(Y=k)=pq^(k-1),k=1,2,3,4,5....compute for E(2Y)Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. FInd (d) fZ|XY (z|x,y), (e) fX|YZ(x|y, z).
- Suppose that two random variables X and Y have the joint PDFfXY(u, v) = {60(u^2)v u ≥ 0, v ≥ 0, and u + v ≤ 1, 0 otherwise.(a) Are X and Y independent?(b) What is the marginal distribution fX(t)?(c) What is Pr[X ≥ Y]? (To set up the right integral, it might help you to draw the rangeof (X, Y) in the uv-plane and identify the region within that range where u ≥ v.)A metal rod of length L and a sphere of radius R are to be joined to form a small automobile part. L and R are normally distributed with the following parameters: E(L) = 2cm, E(R) = 0.5cm, V(:) =0.1^2 cm, V(R)=0.05^2 cm^2. Assume that L and R are independent. What is the probability that the total length of a single assembly is less than 2.90 cm?Let X1, . . . , Xn i.i.d. U([θ1, θ2]), i.e., X1, . . . , Xn are independent and follow a uniform distribution on the interval [θ1, θ2] for θ1, θ2 ∈ R and θ1 < θ2. Find an estimator for θ1 and θ2 using the method of moments.
- Suppose that the random variables X1,...,Xn form a random sample of size n from the uniform distribution on the interval [0, 1]. Let Y1 = min{X1,. . .,Xn}, and let Yn = max{X1,...,Xn}. Find E(Y1) and E(Yn).Let Xi be arandom sample from U(0,1)prove that Xn’ convarges in probability to 0.501)Let x be a uniform random variable over the interval (0, 1). Knowing that y = x2 , calculate:a)Determine Fy(Y) = P(y<=Y),Y real and determine the pdf of y.b)Calculate E[x2] , using the pdf of x.c)Calculate E[y], using the pdf of y and compare with part (b).
- Let X1, .... Xn be a random sample from a population with location pdf f(x-Q). Show that the order statistics, T(X1, ...., Xn) = (X(1), ... X(n)) are a sufficient statistics for Q and no further reduction is possible?If we know that the firm’s profit, X, is uniformly distributed between losing 0.5million dollars to profiting 0.5 million dollars,(a) What is the average?(b) What is the P(losing 0.2 million dollar < x < profiting 0.1 million dollar)?Given that Y1, Y2, Y3, ..., Yn is a random sample from a gamma distribution with parameters alfa = 3, and Beta = theta, find the mle of theta.