Chapters: Normal and Exponential random variables.Q: There exist naturally occurring random variables thatare neither discrete nor continuous. Suppose a group ofpeople is waiting for one more person to arrive beforestarting a meeting. Suppose that the arrival time of theperson is exponential with mean 4 minutes, and thatthe meeting will start either when the person arrives orafter 5 minutes, whichever comes first. Let X denotethe length of time the group waits before starting themeeting.- a. Find P(0

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Answered: :There exist naturally occurring 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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Consider a random walk on {0, ..., k}, which moves left and right with respec- tive probabilities q and p. If the walk is at 0 it transitions to 1 on the next step. If the walk is at k it transitions to k – 1 on the next step. This is called random walk with reflecting boundaries. Assume that k = 3, q = 1/4, p = 3/4, and the initial distribution is uniform. For the following, use technology if needed. 1p. (a) Exhibit the transition matrix.
Suppose #7 buses arrive at a bus stop according to a Poisson process with an average of 5 buses per hour. (i.e. X = 5/hr. So 0=hr. or 12 min.) Find the probability (a) you have to wait longer than 15 minutes for a bus (b) you have to wait more than 15 minutes longer, having already been waiting for 6 minutes.
. A twig of length 1 is broken into two pieces at random, and one piece is thrown away. Theremaining piece is then also broken at random, and one piece of that is thrown away. Let X be thelength of the twig that remains. What is the probability that X > 1/2?
Suppose each of two boxes contains two identical balls marked with numbers 1 and 2. Two balls are selected at random, one from each box simultaneously. Let X denotes the maximum of the numberson the balls selected and Y the minimum of the numbers on the balls selected. Determine the joint probability distribution of X anY
Suppose that customers arrive in a small store according to a Poisson process with an average rate of 1 customer every 5 minutes. Find the probability that, on a given day, the very first customer shows up after 10 minutes.
6.15) Consider a sequence of independent trials, with each trial being a success with probability p. Given that the kth success occurs on trial n, show that all possible outcomes of the first n 1 trials that consist of k 1 successes and n -k failures are equally likely. -
2. Let the random variable X denote a natural number selected at random from 1 to 25 inclusive. Find E[X). а. 12 ь. 12.5 c. 13 d. 13.5 е. 14
Q. 8. What is the chance that a. leap year selected at random wil contain 53 Sundays? [Hint : A leap year consists of 53 complete weeks and 2 days over These two days can be] (1) Monday & Tuesday. (2) Tuesday & Wednesday (3) Wednesday & Thursday (4) Thursday & Friday (5) Friday & Saturday (6) Saturday & Sunday (7) Sundary & Monday Hence P= 2/7
2. Suppose alpha particle emissions of carbon-14 are counted by a Geiger counter each second. The emissions follow a Poisson process with the mean of 16 particles per second. Let W equal the time in seconds before the seventh particle is counted. a) Give the distribution of W. b) Find P(W ≤ 0.5).
Suppose that customers arrive at a diner at a Poisson rate of A = 3 per hour. What is the probability that exactly 6 customers arrive between 1PM and 3PM and that exactly 4 customers arrive between 4PM and 5PM? Hint: You're dealing with two discrete rv's, in this case. (Answer as a decimal number, and round to 3 decimal places).
6. Let X₁ be a r.v. having a Poisson distribution with mean 4, X₂be a random variable having a Poisson distribution with mean 2. What is the distribution of X₁+X₂ if both of the variables are independent? Show this by using MGF's. Find P(X₁+X227).
= Suppose that independent Bernoulli trials, each having probability p 0.4, of being a success are performed until a total of r = 3 success is achieved. Let X be the number of trials required. a) What are the possible values for X? b) IfX = n, find P (X = n). c) What is the probability of having 3 success before having 5 failures?

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