By considering the Theorem of Total Probability and the partitioning resulty-1(Y = y) = U ((Yı = t) n (Y2 = y – t))t=1for y = 2,3, ..., derive the pmf of Y. Suppose that two random variables Y1 and Y, are independent, that is, for all values of y, and y2,P((Yı = y1) n (Y2 = y2)) = P(Y1 = y1)P(Y2 = y2)that is, the events (Yı = y1) and (Y2 = y2) are independent. Suppose also that Yı and Y, have thesame Geometric distribution, that is Y1random variable Y as the sum of Yı and Y2, that is, Y = Y1 +Y2.Geometric(p) and Ý2 ~ Geometric(p). Define a third

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Answered: Suppose that two random variables Y1…

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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Suppose X, Y, and Z are three independent normal random variables. X has an expected value of -3 and a standard deviation of 2.5; Y has an expected value of 4 and a standard deviation of 3.1; Z has an expected value of -1 and a standard deviation of 1.1. Let T = 1X - 2Y + 3Z Variable Mean(Expected Value) Standard Deviation Coefficient a) What is the expected value of T? b) What is the variance of Y? c) What is the variance of T? d) What is the standard deviation of T? X f) What is the probability that T is >0? -3 2.5 1 Y e) What is the probability that T is within 2.0 standard deviations of its mean? Z 4 -1 3.1 1.1 -2 3 k) What is the probability that X + Y> 0? g) What is the probability that T is < -20? h) What is the probability that -20 < T < 0? | i) What is the 25th percentile of the distribution of T? j) What is the probability that the X, Y, and Z are all less than their respective 25th percentiles? Enter a number. 1) Copy your R script or any other comments for the above into…
2. Let X and Y be independent and identically distributed random variables from Uniform(0,1) distribution and Z =. Calculate P(Z < 1.25)
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).
9. For a random experiment with three possible outcomes A1, A2, A3 which of the following gives a legitimate probability assignment? (a) P( A1 )=1/2, P( A2)=1/3, P(A3)= 1/3. (b) P( A1)=1/2, P(A2)=1/6, P(A3 )= 1/6. (c) P( A1)-1/2, P( A2)-2/5, P(A; )- 1/10. (d) P( A1)=1/2, P( A2 )=3/4, P(A3 )= -1/4.
Let X1 and X2 are each independent Bernoulli random variable with probability of success p=0.2 and Y= X1 -3 X22. Compute the expected value of Y. Let X1 and X2 are each independent Exponential random variable with mean 5 and Y= X1 -3 X2 then Compute the value of variance of Y. Let X has Poisson distribution with mean 10. Compute correlation between X and -
23. Let Y denote a geometric random variable with probability of success p. (a) Show that for a positive integer a, P(Y > a) = (1 − p)a (b) Showthatforapositiveintegeraandb,P(Y >a+b|Y >a)=(1−p)a =P(Y >b). (c) Why do you think the property in (b) is called the memoryless property of the geometric distribution? (d) In the development of the distribution of the geometric random variable, we assumed that the experiment consisted of conducting identical and independent trials until the first success was observed. In light of these assumptions, why is the result in part (b) “obvious”? (e) Show that P (Y = an odd integer) = p 1−q2 24. Given that the random variable W is binomial distribution with n trials and success probability p in each trial and P (W = w) = h(w), show that (a) h(w) =p(n−w+1),w>0 h(w − 1) (1 − p)w (b) E [W(W − 1)]= n(n − 1)p2. ? 1 ? 1−(1−p)n+1 (c)E W+1 = (n+1)p
Let Y be a binomial random variable with n = 10 and p = 0.3. (a) P(3 < Y < 5) = P(3 ≤ Y < 5) = (b) P(3 < Y ≤ 5) = P(3 ≤ Y ≤ 5) =
The random variable X takes values -1, 0, 1 with probabilities 1/8, 3/8, 4/8 respectively. a) Write the CDF of X. b) Write the PMF of Y = X² + 2. %3D c) Compute E(Y).
Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)
A bank has 3 counter desks and a very large waiting lounge (assume infinite queue). Customers arrive with a mean of 20 per hour and each desk serves with mean 10 customers per hour, where both rates are exponential. Determine the following : (a) Determine An, tn, C, and Aeff for n = 0, 1, 2, 3... (b) The steady-state probabilities pn,n = 0, 1, 2, 3.. (c) Calculate steady-state measures Ls, Ws, Lg, Wq, c and utilization ratio.
13. Suppose that X and Y are random variables such that Var(X)= 9, Var(Y) = 4, and p(X, Y)= -1/6. Determine (a) Var(X+ Y) and (b) Var(X – 3Y +4). %3D inble

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