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1. Let X and Y be jointly continuous random variables with joint pdf fxy(x, y) = (exp(-2x − y) + exp(-x-2y) x,y>0 - 0 otherwise a. Find the MAP estimate of X given Y = y. b. Find the ML estimate of X given Y = y.

1. Let X and Y be jointly continuous random variables with joint pdf fxy(x, y) = (exp(-2x − y) + exp(-x-2y) x,y>0 - 0 otherwise a. Find the MAP estimate of X given Y = y. b. Find the ML estimate of X given Y = y.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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**1. Let \( X \) and \( Y \) be jointly continuous random variables with joint pdf** \[ f_{X,Y}(x,y) = \begin{cases} \exp(-2x - y) + \exp(-x - 2y) & \text{if } x,y > 0 \\ 0 & \text{otherwise} \end{cases} \] **a. Find the MAP estimate of \( X \) given \( Y = y \).** **b. Find the ML estimate of \( X \) given \( Y = y \).** *You may use the fact that* \[ \int x \cdot e^{-ax} \, dx = \frac{(-1 - ax) \cdot e^{-ax}}{a^2} + C. \]
Transcribed Image Text:**1. Let \( X \) and \( Y \) be jointly continuous random variables with joint pdf** \[ f_{X,Y}(x,y) = \begin{cases} \exp(-2x - y) + \exp(-x - 2y) & \text{if } x,y > 0 \\ 0 & \text{otherwise} \end{cases} \] **a. Find the MAP estimate of \( X \) given \( Y = y \).** **b. Find the ML estimate of \( X \) given \( Y = y \).** *You may use the fact that* \[ \int x \cdot e^{-ax} \, dx = \frac{(-1 - ax) \cdot e^{-ax}}{a^2} + C. \]
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