Q 8.1. Suppose that X and Y are two random variables whose joint distribution is the standard Gaussian distribution in the plane. Let random variables R≥ 0 and = [0, 2π) satisfy X = R cos and Y = Rsin 0. (a) State the density of the joint distribution of R and O. (You do not need to derive this) (b) Express Z = in terms of and hence, making reference to part (a), calculate the density of the distribution of Z. It will help to try to picture this one.
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for part b, could u please provide more details of the calculations I circled? thanks
- 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-Y2Suppose 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)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.)
- The extent to which a distribution is peaked or flat, also called the kurtosis of the distribution, is often mea-sured by means of the quantity α4 = μ4σ4 Use the formula for μ4 obtained in Exercise 25 to findα4 for each of the following symmetrical distributions,of which the first is more peaked (narrow humped) thanthe second:(a) f(−3) = 0.06, f(−2) = 0.09, f(−1) = 0.10, f(0) =0.50, f(1) = 0.10, f(2) = 0.09, and f(3) = 0.06;(b) f(−3) = 0.04, f(−2) = 0.11, f(−1) = 0.20, f(0) =0.30, f(1) = 0.20, f(2) = 0.11, and f(3) = 0.04.We have a random variable X and Y that jave the joint pdf f(x) = {1 0<x<1, 0<y<1} {0 otherwise} If U = Y-X2 , what is the support for the random variable U? What would fu(u) and Fu(u) be? Say U = Y/X, what is the support for the random variable U? What would fu(u) and Fu(u) be?Consider two independent random variables X1 andX2 having the same Cauchy distributionf(x) = 1π(1 + x2)for − q < x < qFind the probability density of Y1 = X1 + X2 by usingTheorem 1 to determine the joint probability density ofX1 and Y1 and then integrating out x1. Also, identify thedistribution of Y1.
- We have a random variable X and Y that jave the joint pdf f(x) = {1 0<x<1, 0<y<1} {0 otherwise} Let U = Y - X2. What is the support for the random variable U? Are there critical points? If U = Y/X. What is the support for the random variable U? Are there critical points?Let Y be a continuous random variable. Let c be a constant. PROVE Var (Y) = E (Y2) - E (Y)2For a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)
- Suppose that Y1, . . . , Yn is a random sample from a population whose density function isAssuming X and Y have joint density f(x; y) = 1/4 for all 0<=x<=2 and 0<=y<=2, and f(x, y) = 0 otherwise.(i) Show that X and Y are independent random variables, and each has the uniform distribution on [0; 2]?(ii) what is E(X) and Var(X)?(iii) Compute Var(X + Y ) and Cov(X + Y, Y )?Let X and Y be discrete random variables with joint pdf f(x,y) given by the following table: y = 1 y = 2 y = 3 x = 1 0.1 0.2 0 x = 2 0 0.167 0.4 x = 3 0.067 0.022 0.033 Find the marginal pdf’s of X and Y. Are X and Y independent?