A First Course in Probability
9th Edition
ISBN: 9780321794772
Author: Sheldon Ross
Publisher: PEARSON
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Chapter 10, Problem 10.8P
(a)
To determine
To Find: The way to simulate from
(b)
To determine
To Find: The way to simulate from
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B1 Suppose X1, ..., Xn
fx(x), where
2
fx(x) = x exp(−x²/0),
0<< (0 otherwise).
(a) Find the maximum likelihood estimator of 0.
(b) Show that the MLE is an unbiased estimator of 0.
(c) Find the MSE of the MLE.
Hint: For parts (b) and (c), you may use integration by parts.
2) Suppose we select two values x and y independently from the uniform distribution on
[0,1]. What is the probability that xy
100 identical balls are rolling along a straight line. They all have speed equal to v, but some of them might move in opposite directions. When two of them collide they immediately switch their direction and keep the speed v. What is the maximum number of collisions that can happen?
Let f(w) be a function of vector w Є RN, i.e. f(w) = 1+e Determine the first derivative and matrix of second derivatives off with respect to w.
Let A Є RN*N be a symmetric, positive definite matrix and bЄ RN a vector. If x ER, evaluate the integral Z(A,b) = e¯xAx+bx dx as a function of A and b.
John throws a fair die with faces labelled 1 to 6. ⚫ He gains 10 points if the die shows 1. ⚫ He gains 1 point if the die shows 2 or 4. • No points are allocated otherwise. Let X be the random variable describing John's gain at each throw. Determine the variance of X.
Chapter 10 Solutions
A First Course in Probability
Ch. 10 - The following algorithm will generate a random...Ch. 10 - Prob. 10.2PCh. 10 - Give a technique for simulating a random variable...Ch. 10 - Present a method for simulating a random variable...Ch. 10 - Use the inverse transformation method to present...Ch. 10 - Give a method for simulating a random variable...Ch. 10 - Let F be the distribution functionF(x)=xn0x1 a....Ch. 10 - Prob. 10.8PCh. 10 - Suppose we have a method for simulating random...Ch. 10 - Prob. 10.10P
Ch. 10 - Use the rejection method with g(x)=1,0x1, to...Ch. 10 - Prob. 10.12PCh. 10 - Prob. 10.13PCh. 10 - Prob. 10.14PCh. 10 - Prob. 10.15PCh. 10 - Let X be a random variable on (0, 1) whose density...Ch. 10 - Prob. 10.1STPECh. 10 - Prob. 10.2STPECh. 10 - Prob. 10.3STPECh. 10 - If X is a normal random variable with mean and...Ch. 10 - Prob. 10.5STPE
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