2) Let X and Y have the joint pmf defined by f(x,y) = where 0 < x < c, 2 < y 4 a. Show that c = 1. b. Show the marginal distributions of X and Y. c. Find P(X > 0.75,Y < 3).
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- 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.)Let X1, X2 denote two independent variables, each with a x^2(2) distribution. Find the joint pdf of Y1=X1 and Y2 = X2+X1. Note that the support of Y1, Y2 is 0<y1<y2<infinity. Also, find the marginal pdf of wach Y1 and Y2. Are Y1 and Y2 independent?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.
- 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?Let X be a r.v. such that E(X2k) = (2k)!/k! , E(X2k+1)=0 . Find the mgf of X and also its ch.f . Then deduce the distribution of X.For the joint distribution function f(x,y)=2 for x≥0,y≥0,x+y≤1 and 0 otherwise Find the marginal of X and Y. Are X and Y independent? Find E(X )
- Let X,Y ~ U(0,1) be iid. Derive the distribution of Z=X/YIf F(x1, x2, x3) is the value of the joint distribution function of X1, X2, and X3 at (x1, x2, x3), show that the joint marginal distribution function of X1 and X3 is given by M(x1, x3) = F(x1, q, x3) for −q < x1 < q, −q < x3 < q and that the marginal distribution function of X1 is given by G(x1) = F(x1, q, q) for −q < x1 < q With reference to Example 19, use these results to find (a) the joint marginal distribution function of X1 and X3; (b) the marginal distribution function of X1.(a∗ ) Compute P(X = 4, Y = 5). (b∗ ) Compute the marginal pmf’s of X and Y . (c∗ ) Are X and Y independent? Let Z be the minimum of X and Y . Calculate E[Z].
- Let x and y be joint continuous random variable with joint pdf f XY (x, y) = {cx+ 1, x, y ≥ 0, x + y < 10, otherwise1. Find the constant c.2. Find the marginal PDF’S fX (x) and fY (y)3. Find P(Y<2X2)Let X1, X2, . . . , Xn be iid from distribution: f(x; θ) = 2x/θ^2 , 0 < x < θ (a) Find a sufficient statistic for θ. (b) Is it minimal?Let x and y be joint continuous random variable with joint pdf f XY (x, y) = { cx+ 1, x, y≥ 0, x+y< 1 0, otherwise 1. Find the constant c. 2. Find the marginal PDF’S fX (x) and fY (y) 3. Find P(Y<2X^2 )
