5.6. Define a collection of events Eq,0 < a < 1, hav-ing the property that P(E)= 1 for all a but= 0.Hint: Let X be uniform over (0, 1) and define eachEa in terms of X.

Given the hint, we will define a random variable X that is uniformly distributed over (0,1).Now,…

Answered: 5.6. Define a collection of events Ea,…

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. Number of cars parked in a parking lot is an example of a O continuous O discrete variable. 4
5. Let S be the set of all finite strings of a's and b's. . Define f : S → Z as follows: For each string s in S, the number of a's in s if s begins with an a f(s): f(s) = the number of b’s in s if s begins with a b O if s = €, that is, if s is the empty word Find the following: (a) f(aba) = (b) f(bbab): (c) What is the range of f? Explain.
Show that the number of planted plane trees on n vertices, m of which are leaves (that is, have zero upward branches) is 1/n (n m) (n-2 m-1) (b) Show that the average number of leaves in planted plane trees onn vertices is n
3. (i) Construct a set S₁ CR so that sup S₁ exists and sup S₁ S₁; (ii) Construct a set S₂ CR so that S₂ is bounded above but not bounded below.
1.47. Which of the following sets are equal? A = {x:x² - 4x + 3 = 0}, B = {x:x²-3x + 2 = 0}, C = (x:xEN, x <3}, D= (x:xEN, x is odd, x < 5}, E = {1,2}, F = {1,2, 1}, G=(3, 1), H = {1, 1,3} 1.48. List the elements of the following sets if the universal set is the English alphabet U = {a, b, c,..., y, z}. Furthermore, identify which of the sets are equal. A = {x:x is a vowel}, B = {x:x is a letter in the word "little"}, C = {x:x precedes f in the alphabet}, D = {x: x is a letter in the word "title"}
2.6. Let (2, F, P) be a probability space and let A1, A2,... be sets in F such that P(Ag) < ∞ . k=1 Prove the Borel-Cantelli lemma: P(N U Ak) = 0, m=1 k=m i.e. the probability that w belongs to infinitely many A{s is zero.
5 Consider H, whose preferences over lotteries with only two prizes ($100, $0) are as follows: H strictly prefers lottery L to lottery L’ iff p > p’ where p and p’ (respectively denote) the probability of $100 prize in lotteries L and L’. Does H’s preferences over lotteries on ($100, $0) satisfy continuity, independence? (Do not just answer Yes, or No. If you say Yes, clearly show why. If you say No, then provide a counterexample.)
Q4. Prove that for any three events A, B and C (Inclusion-Exclusion Principle). P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
1. Let S = (0,7) U 27 +13... Define < on S by a
4
The duration, x, of a monthly faculty meeting is uniformly distributed between 30 and 80 minutes. Which of the following statements is true (check all that apply)? a. Prob (x>80) = 0 b. Prob (x<85) = 1 c. Prob (x>35) = 1 d. Prob (x<30) < 0
A certain market has both an express checkout line and a superexpress checkout line. Let X, denote the number of customers in line at the express checkout at a particular time of day, and let X, denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of X, and X, is as given in the accompanying table. X2 1 3 0.09 0.07 0.04 0.00 1 0.05 0.15 0.05 0.04 X1 0.05 0.04 0.10 0.06 0.01 0.02 0.04 0.07 4 0.00 0.02 0.05 0.05 (a) What is P(X, = 1, X, = 1), that is, the probability that there is exactly one customer in each line? P(X, = 1, X, = 1) = | (b) What is P(X, = x,), that is, the probability that the numbers of customers in the two lines are identical? P(X, = X,) = (c) Let A denote the event that there are at least two more customers in one line than in the other line. Express A in terms of X, and X,. O A = {X, 2 2 + X2U X, 2 2 + Xq} O A = {X, s 2 + X2 U Xq 2 2 + Xq} O A = {X, s 2 + X2 U Xq S2+ Xq} O A = {X, 2 2 + X2 U Xq S2+ Xq} Calculate…

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5.6. Define a collection of events Ea, 0 < a < 1, hav- ing the property that P(E)= 1 for all a but P(E)=0. Hint: Let X be uniform over (0, 1) and define each Ea in terms of X.