5. Suppose that X and Y are two independent discrete random variables, where X and Y onlytake positive integer values (1, 2, 3, 4.). Suppose P(X < 1/2) = , P(X > 3/2) =÷, andP(Y = 1) = Compute the probability P(X = 1nY = 1).%3D

Independent events: Two or more events are said to be independent if occurrence of one does not…

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A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

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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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Let X1, X2,..., X81 be i.i.d. (independent and identically distributed), each with expected value = E(Xi) using the central limit theorem. 5, and variance o? = Var(Xi) 4. Approximate P(Xı + X2 + · .. X81 > 369),
8. Let X1, X2 be i.i.d. random variables from a distribution with p.d.f fx(x) = otherwise and let Y = X(1) = min(X1, X2). Calculate P(1
A psychologist determined that the number of sessions required to obtain the trust of a new patient is either 1, 2, 3, 4, or 5. Let X be a random variable indicating the number of sessions required to obtain the trust of a new patient and the probability function of X is f (x)= (6 - x)/15 for x=1, 2, 3, 4, or 5. What is the mean of the number of sessions required to obtain the trust of a new patient? 單選: A. 2.33 В. 3.33 C. 3.67 D. 3.85 E. 2.67
2. Let Y,,., Y, be independent random variables such that Y, (Yı., Yp)" and 0 = (0,.,0p)". Let = 0(Y) = (0,(Y),... , @p(Y))" be an estimator of 0, and let g(Y) = (g(Y),... , gp(Y))" = – Y. Denote by || - || the Euclidean norm, ||Y° = Y} + .. + Y. N(8), 1). Write Y = %3D Suppose that D(Y) = @g(Y)/ay, exists. Then it is known that %3D R(Ô(Y)} = +2 D(Y) + É19(Y)² =1 is an unbiased estimator of the risk of 0, under squared error loss L(0, ê) = ||0 – e|P. [You are not required to show this]. %3D (i) The James-Stein estimator is 6.s(Y) = (1 – )Y. _P-2y ||Y? Show that an unbiased estimator of the risk of d Js(Y) is Řlójs(Y)) = p – (p - 2) /||YII°. Deduce that Y is inadmissible as an estimator of 0. Is ô js(Y) admissible? Justify your answer.
A sample is selected from one of two populations, S, and S,, with P(S,) = 0.9 and P(S,) = 0.1. The probabilities that %3D 1 2' an event A occurs, given that event S, or S, has occurred are P(A|S,) = 0.4 and P(A|S2) = 0.3 %3D and the probability of event A is P(A) = 0.39. Use Bayes' Rule to find P(S, JA). (Round your answer to four decimal places.)
Suppose that X is a geometric random variable with parameter π. Show that P(X = n+k|X > n) = P(X = k).
10. Let and be independent random variables representing the lifetime (in 100 hours) of Type A and Type B light bulbs, respectively. Both variables have exponential distributions, and the mean of X is 2 and the mean of Y is 3. a) Find the joint pdf f(x, y) of X and Y. b) Find the conditional pdf f₂ (ylx) of Y.. c) Find the probability that a Type A bulb lasts at least 300 hours and a Type B bulb lasts at least 400 hours. d) Given that a Type B bulb fails at 300 hours, find the probability that a Type A bulb lasts longer than 300 hours. e) What is the expected total lifetime of two Type A bulbs and one Type B bulb?
Suppose X1, X2, X3 and X4 are independent random variables, which havethe same Bernoulli distribution with parameter p = 1/3. Compute Cov(X1 + X2 +X3, X2 + X3 + X4)
If X and Y are independent random variables with X~N(2,32) and Y~N(1,42),find P(X<Y). (You need to write out the calculation process and figure out theanswer)
2. Suppose that Y is a binomial random variable based on n trials with success probability p and consider Y* = n - Y. a Argue that for y* = 0, 1, ..., n P(Y* = y*) = P(n − Y = y*) = P(Y = n − y*). b Use the result from part (a) to show that n P(Y* = y*) = · ( ₁₁² y.)pª²-xq" = ("₁. )¶" p²-x. - y* c The result in part (b) implies that Y* has a binomial distribution based on n trials and "success" probability p* = q = 1 – p. Why is this result "obvious"?
Exercise 4. Let = {a,b,c,d} and define a probability measure on 222 by P(a) = P(c) = 1/6 and P(b) = P(d) = 1/3. Define random variables X and Y by X(a) = X(b) = 2, X(c) = X(d) = −2 Y(a) = Y(d) = -1, Y(b) = Y(c) = 1. (a) List all sets in σ(Y). (b) Let Z = E(XY), determine Z(a) Z(b), Z(c) and Z(d).
d,e,f

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5. Suppose that X and Y are two independent discrete random variables, where X and Y only take positive integer values (1, 2, 3, 4.). Suppose P(X < 1/2) = , P(X > 3/2) =÷, and P(Y = 1) = Compute the probability P(X = 1nY = 1). %3D