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.)
The mean (or expected value) of a probability distribution can be calculated by first making the probability distribution. Then multiply the possible values of the Random Variable by their probabilities (X*P(X)) then add all the X*P(X) up. On the TI 83/84 calculator you can 1) Enter the values of the RV's into List 1 (Press STAT, 1 (for Edit), highlist L1 and press CLEAR if needed, then enter values into the list) 2) Enter the probabilities into L2 (press > and then enter the values in the list. highlight L2 and press CLEAR first if needed). 3) 2nd MODE (to quit the list) 4) STAT, > (for CALC), select 1-Var Stats, List: L1 (2nd 1), FreqList: L2 (2nd 2). Then calculate. x¯ is the expected value (mean) of the frequency distribution. A game is played. If you win you gain $4. If you lose you lose $10. The probability of winning is 0.747. What is the expected value (mean)?
When computer generates random digits, the probability that any particular digit in the set {0,1,2,... 9}is generated on any individual trial is 1/10 0.1. Suppose that we are generating digits one at a time and let X the number of digits that the computer generates to get the first zero. (a) Detemine the probability that the first 0 occurs on the fifth random digit generated_ (6) How many random digits would you expect to have to generate in order to observe the first 0? (c) Calculate and interpret the standard deviation of X.
4.1 Let X have a geometric distribution, that is p(x) = q*-'p x = 1,2,3, .. Where X has the memoryless property given by: P(X > k + b|X > k) = P(X > b). Given that you have already tossed a balanced coin 5 times and obtained Zero heads, what is the probability that you must toss it at least 3 more Times to obtain the first head?
let Xxx Xz A randomsample that follows a normat exponential ExpC@), The Probability distribution '- ExP(no) 2 -ExP (n/0) Not in Ahe answer
The flow of goods through a certain machine in a factory can be modeled as a Poisson Random Process. Suppose that a machine has 20 goods per minute as an outflow rate, determine the probability that the 7th good will exit within 30 seconds. (Express your answer in 4 decimal places)
b) Suppose that the number of calls at a switch board during a 30 minute time interval is known to be a random variable having a poisson probability distribution with parameter ?=3. Find the probability that during a given 30 minutes,?2i) Exactly 4 calls will be receivedii) More than 4 calls will be received iii) At least one call will be received
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
According the World Bank, only 8% of the population of Uganda had access to electricity as of 2009. Suppose we randomly sample 20 people in Uganda. Let X = the number of people who have access to electricity. The distribution is a binomial. What is the distribution of X? X ~ ? B U N (,) 2. What is the probability that between 3 and 6 (including 3 and 6) people have access to electricity in this study?
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
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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