Problem 4 (counts as two problems): Let X1, X2, ., X„be a collection of independent discrete random variablesthat all take the value 1 with probability p and take the value 0 with probability (1-p). The following set of stepsillustrates the Law of Large Numbers at work.Compute the mean andfV (which infô-tcx, +xomnute themaan nved in theninatnses of X..c) Suppose n = 10,000. Use Chebyshev's inequality to provide an upper bound for the probability that thedifference between p and p exceeds 0.05.

Bernoulli distribution: A random variable is said to have a Bernoulli distribution with parameter p…

Answered: Problem 4 (counts as two problems): Let…

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.)
5. A candy dish contains 2 red (R) and 4 yellow (Y) candies. You close your eyes, select two candies from the dish (without replacement), and record their colors. Use general multiplication rule and special addition rule to find the following probabilities. a. The probability that the first selected candy will be red and the second selected candy will be yellow is b. The probability that two selected candies will be of different colors_ One is red, another one is yellow c. Probability that two selected candies will be of the same color_ Both are red or both are yellow Assuming that the event "The first selected candy will be red and the second selected candy will be yellow "is denoted by A, the event "Two selected candies will be of different colors" is denoted by B, and the event "Two selected candies will be of the same color" is denoted by C, show your work in finding P(A), P(B), and P(C) in the questions a, b, and c.
Before every flight, the pilot must verify that the total weight of the load is less than the maximum allowable load for the aircraft. The aircraft can carry 43 passengers, and a flight has fuel and baggage that allows for a total passenger load of 7,181 Ib. The pilot 7,181 lb sees that the plane is full and all passengers are men. The aircraft will be overloaded if the mean weight of the passengers is greater than - = 167 Ib. What is the probability that the aircraft is overloaded? Should the pilot take any action to correct for an 43 overloaded aircraft? Assume that weights of men are normally distributed with a mean of 178.9 lb and a standard deviation of 35.4. The probability is approximately (Round to four decimal places as needed.)
Consider two mutually exclusive events A and B such that P((A or B)^c)=4/7 and P(B)=1/7. Then P(A^C) is equal to Note: P(A^C) means probability of A complement О РАor B)+P(A) ЗР(В)+0.5 O P(B^C) O 2P(A)
Before every flight, the pilot must verify that the total weight of the load is less than the maximum allowable load passengers, and a flight has fuel and baggage that allows for a total passenger load of 5,670 lb. The pilot sees that the plane is full and all passengers are the aircraft. The aircraft can carry 35 5,670 lb men. The aircraft will be overloaded if the mean weight of the passengers is greater than = 162 lb. What is the probability that the aircraft is 35 overloaded? Should the pilot take any action to correct for an overloaded aircraft? Assume that weights of men are normally distributed with a mean of 182.8 lb and a standard deviation of 39.3. The probability is approximately. (Round to four decimal places as needed.) Should the pilot take any action to correct for an overloaded aircraft? O A. Yes. Because the probability is high, the pilot should take action by somehow reducing the weight of the aircraft. O B. No. Because the probability is high, the aircraft is safe to…
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?
The items produced by a manufacturer have a probability equal to 0.3 of being defective. For a random sample of 10 items, the probability that at most 3 items are defective as given by R is 1-pbinom(3,10,0.3) pbinom (10,3,0.03) pbinom(3,10,0.3) dbinom(3,10,0.3)
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).
Q4) ( / In a laboratory two types of tests done for a concrete cubes: compression streng.h test and water absorption test .40% passed in compression test. 20% failed in water absorption test and 12% failed in both. Find: 1) the probability that a random sample selected is bad in compression test and good in absorption test. 2) the probability that a random sample selected is bad in absorption test and good in compression test. 3) the probability that a random sample selected is good in both test. 4) the probability that a random sample selected at least one of them is bad.
1. Given that X~N(39,32)A Random variable X is normally distributed and has a mean of 9 C Random variable X is not normally distributed and the mean is 39 B Random variable X is normally distributed and has a mean of 3 D Random variable X is normally distributed and the mean is 39
4. The distribution function of a random variable W is given by: Oforw 2 a. Find P (1 < w< 3). Enter probability as a reduced fraction. prob = %3D b. Find f(w). Enter coefficients as reduced fractions and express powers with ^. f(w) 3= for -2< w < 2 %3D
3. An urn contains exactly 10 red chips and 6 blue chips. Five chips are randomly selected without replacement. Let X denote the number of blue chips in the sample of five chips. (a) What type of random variable is X? Indicate both the type and the appropriate parameters using the "~" notation. Write down the pmf of X, and do not forget to indicate the range of values that x can take on. X ~ px (x) (b) What is the probability that there are between two and four blue chips among the five chosen chips? Express your answer as a decimal. = (*)*

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