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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...
Suppose X is an exponential random variable with PDF, fx(x) exp(-x)u(x). Find a transformation, Y g(X) so that the new random variable Y has a Cauchy PDF given 1/π . Hint: Use the results of Exercise 4.44. ) Suppose a random variable has some PDF given by ). Find a function g(x) such that Y g(x) is a uniform random variable over the interval (0, 1). Next, suppose that X is a uniform random variable. Find a function g(x) such...
A fixed speed wind turbine can be characterized by the following equation A fixed speed wind turbine can be characterized by the following equation rpm and R is the length of the blade -37 m. The wind speed is a random variable with the following CDF and PDF The characteristic wind speed is -30 m/s. Determine the following: (a) Find the PDF of G, fG(g), using the solution in section 1.3.1 of the notes (b) Find the equation for the...
Let Y = g(X) = , where the PDF is a) Determine b) Determine c) Determine PDF of d) Determine the CDF of e) What is the
Problem 2. Assume a random vector (X Y with cdf F(r, ) and pdf f(r,y) (i) Show that Y/X has the pdf f(x, z) |da, g(z) = (ii) For X and identify the distribution of this pdf. xt independent, evaluate the pdf of Y/VX N(0, 1) and Y
Only 1-3) ,X, be a random sample from N(u,0") and let X and S be sample 1. Let mean and sample variance, respectively. In order to show that X and S are independent, tollow the steps below. x - x -X, and show the joint pdf of ,X,,..., X 1-1) Use the change of variable technique is (n-1)s n-u) еxp f(X,x 2a 20 av2n Use Jacobian for n x n variable transformation 1-2) Use the fact that X~N(4, /n), and...
Only 1-3) ,X, be a random sample from N(u,0") and let X and S be sample 1. Let mean and sample variance, respectively. In order to show that X and S are independent, tollow the steps below. x - x -X, and show the joint pdf of ,X,,..., X 1-1) Use the change of variable technique is (n-1)s n-u) еxp f(X,x 2a 20 av2n Use Jacobian for n x n variable transformation 1-2) Use the fact that X~N(4, /n), and...
D Question 3 10 pts The joint pdf of X, Y, and Z is given by f(x,y,z) = 8xyez for 0sxcys 1, z>0 (0 otherwise) What is the joint pdf of U=X/Y and V = Y? O g (u, u) = 8uu for 0 < u-u-1 (-0 otherunse) O g(u,) foru 1 0v1 (0 otherwise) O g (u, u) 4uv for 0 < u < 1; 0 < u < 1 ( 0 otherwise) O g (u, v) = 8uv2...
2. The actual tracking weight of a stereo cartridge that can be set to track at 3 g on a particular changer can be regarded as a continuous random variable X with pdf fx(x) otherwise a. Find the value of k so that fx is a valid pdf. b What is the probability that the actual tracking weight (X) is greater than the prescribed weight of 3 g? c. What is the probability that the actual weight is between 2.75...
Problem # 8. a) Let X be a continuous random variable with known CDF FX(x). LetY = g(X) where g(·) is the so-called signum function, which extracts the sign of its argument. In other words, g(X) = { -1 x<0, 0 x=0, 1 x>0 } Express the PDF fY (y) in terms of the known CDF FX(x). b) Let X be a random variable with PDF: fX(x) = { x/2 0 <= x < 2, 0 otherwise} Let Y be...
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