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)

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Author:Sheldon Ross

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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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Suppose X, Y are two discrete random variables with probability function given by: Y \ X |1 -1 -1 1/9 2/9 1/9 2/9 1/9 1/9 1/9 Then X and Y are: Independent Dependent Question * The time taken by a student to finish a statistics exam can be modeled using the following cumulative distribution function: F(x) = 1-e-0.9x, x > 0 Suppose that the time taken for a student to finish the exam is constant from student to student. Then the mean number (to the nearest whole number) of students who will finish the exam in less than 1 hour for 12 students randomly selected is equal to: None of the other options O 7 8.
b) A random variable is normally distributed with u = 50 ando² = 25. Find the probability (i) that it will fall between 55 and 100, (ii) that it will be greater than 54.
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
Let X, and X, be independent random variables. Suppose the mean of X, and X2 are 2.5 and 3; and variance of X, and X2 are 2 and 3, respectively. 1) E (3 X +3.5 X,) = ; 2) Var (3 X +3.5 X,) = ;
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.)
Prove the following
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)

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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