PROBLEM 4 Let X be a continuous random variable with the following PDF 6x(1 - 1) if 0 <r<1 fx(x) = o.w. Suppose that we know Y X = ~ Geometric(2). Find the posterior density of X given Y = 2, i.e., fxy (2/2).
1. Let X be a continuous random variable with the probability density function fx(x) = 0 35x57, zero elsewhere. Let Y be a Uniform (3, 7) random variable. Suppose that X and Y are independent. Find the probability distribution of W = X+Y.
Let X be a continuous random variable with the following PDF 6x(1 – x) if 0 < x < 1 fx(x) = 3 0.w. Suppose that we know Y | X = x ~ Geometric(x). Find the posterior density of X given Y = 2, i.e., fxY (2|2).
Problem 4 Let X be a discrete random variable with probability mass function fx(x), and let t be a function. Define Y = t(X): that is, Y is the randon variable obtained by applying the function t to the value of X Transforming a random variable in this way is frequently done in statistics. In what follows, let R(X) denote the possible values of X and let R(Y) denote the possible values of To compute E[Y], we could irst find...
STAT 115 Let X be a continuous random variable having the CDF Fx(x) = 1 - e^ (-e^x) (1) Find the Probability Density Function (PDF) of Y=e^X. (2) Let B have a uniform distribution over (0,1). Find a function G(b) and G(B) has the same distribution as X.
Question 3: Let X be a continuous random variable with cumulative distribution function FX (x) = P (X ≤ x). Let Y = FX (x). Find the probability density function and the cumulative distribution function of Y . Question 3: Let X be a continuous random variable with cumulative distribution function FX(x) = P(X-x). Let Y = FX (x). Find the probability density function and the cumulative distribution function of Y
Let X be a random variable with density function fX (x)= cx(1−x), if0<x<1, 0 ,otherwise. (a) What is the value of c? (b) What is the cumulative distribution function FX for X? (c) What is the probability that X < 1/4?
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...
Question 5 15 marks] Let X be a random variable with pdf -{ fx(z) = - 0<r<1 (1) 0 :otherwise, Xa, n>2, be iid. random variables with pdf where 0> 0. Let X. X2.... given by (1) (a) Let Ylog X, where X has pdf given by (1). Show that the pdf of Y is Be- otherwise, (b) Show that the log-likelihood given the X, is = n log0+ (0- 1)log X (0 X) Hence show that the maximum likelihood...
Let X be a continuous random variable with PDF fx(x)- 0 otherwise We know that given Xx, the random variable Y is uniformly distributed on [-x,x. 1. Find the joint PDF fx(x, y) 2. Find fyo). 3. Find P(IYI <x3) Let X be a continuous random variable with PDF fx(x)- 0 otherwise We know that given Xx, the random variable Y is uniformly distributed on [-x,x. 1. Find the joint PDF fx(x, y) 2. Find fyo). 3. Find P(IYI