Problem 3. Suppose that X and Y are independent Poisson random variables with param-eters A and μ, respectively. Find the distribution of X+Y. Prove that the conditionaldistribution of X, given that X + Y = n, is binomial with parameters n and λ/(\+µ). (Fortwo random variables Z and T, the conditional distribution of Z given T is given byPz|T(z|t) = P(Z = z |T = t),for all t such that P(T = t) > 0.)

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Answered: roblem 3. Suppose that X and Y are…

A First Course in Probability (10th Edition)

10th Edition

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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 be 2 mutually independent discrete random variables. The first variable X₁ can be a whole number from 1 to 5. Its distribution function is 6 - i m₁ (i) 15 The second variable X₂ can be a whole number from 1 to 6. It has a uniform distribution. First, find the exact values (using fractions) of P(X₁ = 3) = 1/5 : P(X₂ = 5) = 1/6 P(X₁ = 3 and 3 and X₂ = 5) = 1/30 X₂ # Let Y denote the maximum of the X¿'s. Find the probability that Y = 1: P(Y = 1) = 1 Find the probability that Y = 2, P(Y= 2) = 2/45 Hint: Use cases based on what X₁ and X₂ are.
71. Given two independent random Y~ Bin(n,p) with mgf my (t) of X + Y. variables, X~ Bin(m, p) with mgf mx(t) = [pet + (1 - p)], and [pet + (1 p)]", apply Theorem 12.3 to determine the distribution =
4. Let X₁ N (0,3²), X₂ N(4,4²), and X3 ~ N(-1,22) be independent random variables. Find P (X₁ + X₂ +3X3 <-8), using the table for standard normal distribution.
Suppose that the number of traffic accidents, N, in a given period of time is distributed as a Poisson random variable with E(N)=100. Use the normal approximation to the Poisson to find A such that P(100 - A
A company has 6000 arrivals of Internet traffic over a period of 12,160 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these μ*• e -μ to find the probability of X! Internet arrivals have a Poisson distribution. If we want to use the formula P(x) = exactly 3 arrivals in one thousandth of a minute, what are the values of µ, x, and e that would be used in that formula? μ= (Round to three decimal places as needed.) X = e = (Round to three decimal places as needed.) tv -O). ☐☐ + S il 1 √₁ Vi A O (LI) More W Next X
Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)
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Q 6.1. Suppose Z = (Z1, Z2, Z3) is a standard multi-variate Gaussian random variable i.e., for i ≤ 3, Zi~ N(0, 1) are i.i.d. random variables. Each of the random variables, (a)-(d), on the left is equal in distribution to exactly one random variables, (1)-(4), on the right. Pair up according to "equal in distribution" and explain briefly your reasoning. (a) (X₂) (¹) (X) = (2) 1/√2 (Ⓒ) (x₂) - (3/1² ¹1/1²) (2) (c = 2 0 = 2 Z₁ 1) (²) 3 (4¹) (X) = (²¹) (1) (2) (3) Y₁ 1 (29) - ()) (2) = Y₂ 1 Y₁ (121) = (1/² √₂) (2) (1/√2 1/√2 /2 −1/√√2, Y₂ X₁ X₂ = 22 1 1 2 0 Y₁ 2 1 (4) (2)) = (²) (²) 1 1 Z₁ Z₂ Z3

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roblem 3. Suppose that X and Y are independent Poisson random variables with param- ers λ and µ, respectively. Find the distribution of X + Y. Prove that the conditional stribution of X, given that X+Y = n, is binomial with parameters n and λ/(\+µ). (For 1 1:1 71 7 67 1m