2.Suppose that Y is a binomial random variable based on n trials with success probability p andconsider Y* = n - Y.a Argue that for y* = 0, 1, ..., nP(Y* = y*) = P(n − Y = y*) = P(Y = n − y*).b Use the result from part (a) to show thatnP(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"?

a) To show that for , we can use the properties of the binomial distribution.Y is a binomial random…

Answered: 2. Suppose that Y is a binomial random…

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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3. (Competing patterns among coin flips) Suppose that Xn, n > 1 are i.i.d. random variables with P(X1 = 1) = P(X1 = 0) = }. (These are just i.i.d. fair coin flips.) Let A = (a1, a2, a3) = (0, 1, 1), B = (b1, b2, b3) = (0,0, 1). Let TA = min(n 2 3: {Xn-2, Xn-1, Xn) = A} be the first time we see the sequence A appear among the Xn random variables, and define TB similarly for B. Find the probability that P(TA < TB). (This is the probability that THH shows up before TTH in a sequence of fair coin flips.)
Suppose X is a random variable with standard beta distribution and parameters (alpha) = 8 and (beta) = 2 What is (PX>=0.6)?
5
Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)
34. Suppose that X is a random variable with MGF M(t) = (1/8)e + (1/4)e + (5/8)e. (a) What is the distribution of X? (b) What is P[X= 2]?
Let E and F be two events of an experiment with sample space S. Suppose P(E) = 0.46, P(F^) = 0.65, and P(E U F) =0.7. Calculate the probabilities below. (a) P(E) = (b) P(F) (c) P(E n F) =
13) (3 points each) Suppose X and Y are independent random variables each having expected value 5 and standard deviation 3. Calculate d) If you were told that Cov(X,Y) = 4, and that X and Y are not independent, what would Var(X-Y) be?
10. Two random variables X and Y take on the values i and 2 with probability 1/2' (i = 1,2, ...). Show that the probabilities sum to one. Find the expected value of X and Y.
6. A transportation department is considering placing stop lights at a busy 4-way-stop intersection. Let X1, X2, X3, X4 represent the number of drivers arriving per minute at the respective stop signs. Assume- these to be four mutually independent Poisson random variables with respective means of 2, 5, 4 and 2. (a) Find the mean and standard deviation of S = Xị. (b) Confirm that S ~ Poisson(13) (c) Compute P(S = 0).
24. Gaussian random variables X, and X,, for which, X, = 2, o = 9, X, =-1, ox, = 4 and Cxx, =-3, are transformed to new random variables Y, and Y, according to जापर डेरें| Y, =-X, + X, Y, =-2X,- 3X, (iii) Cy,Cy, .. Find
Let x1, x2, .. xn be independent standard normal random variables, where n = 9. What is the probability that 3 of them have values in the range [1, 2]? This is how alpha is computed for some sentizing rules. Use 7 decimal places.
9. Let X₁, X2, ..., X5 be a random sample from a distribution with mean and variance 5 X₁, find Cov(X₁, X) 2. If X= = i=1

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