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1. Consider the uniform distribution X defined over the interval [0, 2pi]. Now let Y = sin(X) (a) Calculate the CDF FY(y) of Y. (b) Calculate the PDF f(y) of Y. In particular, in what interval [a, b] is Y defined? (this mean f(y) = 0 for y < a and for y > b). (c) Verify that f(y) is a PDF.
Question 1: (10 marks) Let Y, Y....,Y, be a random sample from the beta distribution with a = B = 4, and I2 = { u u = 1,2). Write the likelihood ratio test statistic A for testing Ho : H = 1 versus H:u= 2. Note that the pdf of a beta(a,b) distribution is as follows: com_(a+b)/2-1(1 - 0)8-1, 0<I<1. f(x) = f(a)(B)"
Consider the hierarchical Bayes model (a) Show that the conditional pdf g(ply, 0) is the pdf of a beta distribution with parameters (b) Show that the conditional pdf g(θ|y, p) is the pdf of a gamma distribution with parameters 2 and log p Consider the hierarchical Bayes model (a) Show that the conditional pdf g(ply, 0) is the pdf of a beta distribution with parameters (b) Show that the conditional pdf g(θ|y, p) is the pdf of a gamma distribution...
11. Obtain the MLE estimate for the beta parameter in Gamma distribution defined below for n iid (identical and independent) observations in a sample. Show steps. Obtain the MLE estimate for the alpha parameter. The continuous random variable X has a gamma distribution, with param eters α and β, if its density function is given by x>0, elsewhere, .tor"-le-z/ß, f(x; α, β)-Ί where α > 0 and β > 0. (You will also need the beta estimate, use the direct...
Problem 2: 10 points A random variable, Z, has the Gamma distribution with the density: and f ()0, elsewhere. According to the notation in Probability Theory, Z has the distribution Gamma [2. Conditionally, given Zz, a random variable, U, is uniformly distributed over the interval, (0,z) 1. Evaluate the joint density function of the pair, (Z, U). Indicate where this density is positive. 2. Derive the marginal density, fU (u) 3. Find the conditional density of Z, given Uu. Indicate...
Suppose y has a「(1,1) distribution while X given y has the conditional pdf elsewhere 0 Note that both the pdf of Y and the conditional pdf are easy to simulate. (a) Set up the following algorithm to generate a stream of iid observations with pdf fx(x) 1. Generate y ~ fy(y). 2. Generate X~fxy(XY), (b) How would you estimate E[X]? Suppose y has a「(1,1) distribution while X given y has the conditional pdf elsewhere 0 Note that both the pdf...
The following joint probability distribution is given. 1. Find k such that the given function demonstrates the PDF. 2. Find Marginal distributions. 3. Evaluate ?(? < ? < 0) 4. Find the correlation coefficient between X and Y having the joint density functions:(.) ?(?,?) = {???2+?2 ??? ?2 + ?2 < 4 0 ?????h??? Question 2. (20 pts.) The following joint probability distribution is given. 1. Find k such that the given function demonstrates the PDF. 2. Find Marginal distributions....
01. 125 marks Find the approximate distribution of y.. (sample median) when the population has; a) Uniform distribution U(0, b) Exponential distribution Exp(0) c) Beta distribution Beta(α, β) d) Gama distribution Gamma(α,β)
The answer mean is 1/3, variance is 1/18 Problem 44.15 Suppose that X has a continuous distribution with pdf. fx (x) = 2x on (0,1) and 0 elsewhere. Suppose that Y is a continuous random variable such that the conditional distribution of Y given X- is uniform on the interval (0, x). Find the mean and variance of Y.
Please show both joint density function of (X,Y) and the name of the distribution. Exercise 6.17. Let U and V be independent, U Unif(0,1) and V~ Gamma(2, x) which means that V has density function v0 and zero elsewhere. Find the joint density function of (X, Y) (UV, ( 1-U) V). Identify the joint distribution of (X.Y) in terms of named distributions. This exercise and Example 6.44 are special cases of the more general Exercise 6.50. fv (v-λ-e-Av for