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Cumulative Distribution Function, Renewal Function and Poisson

3. Let {N(t)}>0 be a renewal process for which the interarrival times {T1,T2,. . .} have cumulative distribution function F. Recall that the renewal function m(t) is given by m(t) E[N(t)j. (a) Find E[N(t)Ti=x] fort<x andfortx. (b) Find E[N(t)] E[E[N(t) T1]] to prove that m(t) satisfies the renewal equation m(t) F(t) + f m(t

Stochastic Processes : Poisson Process and Markov Chains

1. Suppose that shocks occur according to a Poisson process with rate A> 0. Also suppose that each shock independently causes the system to fail with probability 0 < p < 1. Let N denote the number of shocks that it takes for the system to fail and let T denote the time of the failure. (a) FindP{T>tNrrn}. (b) FindP{NrrnT=t}. (

Probability and Statistics : Tree Diagram

Two friends Dave and Pete often play squash and tennis together. Over the years, they have found that Dave wins 3 out of every 5 rounds of squash and 1 out of every 4 tennis matches. If they play one match of squash and then one tennis match: a) Draw a tree diagram to describe the situation b) Find the probability that Dave

Statistics and Probability

1. Each of the three measures of central tendency-the mean, the median, and the mode-are more appropriate for certain populations than others. For each type of measure, give two additional examples of populations where it would be the most appropriate indication of central tendency. 2. Find the mean, median, and mode of

Statistics : Standard Deviation and Mean (10 Problems)

1. True or False? The standard deviation of a population will always be smaller than the standard deviation of the sample means (of the samples used to estimate this same population). 2. The mean height for a population is 65 inches and the standard deviation is 3 inches. Let X_ denote the mean height for a sample of people

Statistics : Probability of Winning a Powerball Lottery

The Powerball For a single ticket, a player first selects five numbers from the numbers 1-53 and then chooses a powerball number, which can be any number between 1 and 42. A ticket costs $ 1. In the drawing, five white balls are drawn randomly from 53 white balls numbered 1-53, and one red Powerball is drawn randomly from

Probability of Scoring a Hole-In-One in Golf

See the attached files. Recall that the case study in your text concerns an amazing event that occurred during the second round of the 1989 US Open at Oak Hill in Pittsford, New York. Four golfers--Doug Weaver, Mark Wiebe, Jerry Pate, and Nick Price--made holes-in-one on the sixth hole. Recall from your text that the probabi

18 Problems : Payoff Tables and Decision Trees; Control Charts; Bayes Theorem; Total Quality Management ( Demings 14 Points of Management ), Expected Monetary Value, Red Bead Experiment

1. A tabular presentation that shows the outcome for each decision alternative under the various states of nature is called a: a. payback period matrix. b. decision matrix. c. decision tree. d. payoff table. 2. The difference between expected payoff under certainty and expected value of the best act without certainty is

Concepts of probability statistics

1. Event A: You spend your entire Memorial Day 2006 in Acapulco. Event B: You spend your entire Memorial Day 2006 with some of your friends. TRUE OR FALSE: A and B are mutually exclusive events. 2.Event A: You spend your entire Memorial Day 2004 in Denver. Event B: You spend your entire Memorial Day 2004 in Vermont. TRU

Independent random variables

1. A coin is tossed 3 times. Discrete random variable X is equal to the number of times Heads comes up. Discrete random variable Y has the value 1 if the first toss comes up heads and 0 otherwise. (a) Find Pr[(X=1)n(y=1)] Are X and Y independent random variables?

Probability - drawing tokens

7. There are some tokens in a bag. Each token has a number ( a positive integer) printed on it. Ernesto does not know how many tokens are in the bag, and the only thing he knows about the numbers on them is that the mean of the numbers is 3 and the variance is 2. Ernesto is going to play a game in which he draws 1 token from th

Probability and Cumulative Distribution Functions

4. A hand of 5 cards contains 2 red cards and 3 black cards. Trish plays the following game: A card is drawn from the hand. If the card is red, the game stops immediately. If the card is black, this black card is set aside and a red card is put into the hand in its place. Then another card is drawn from the hand and the same pro


1. There are 3 boxes. Each box contains several envelopes. Some envelopes have "you lose" written on them. The rest say "you win..." . However, of the ones that say "you win..." , some contain only a piece of paper saying "... nothing" . Each of the other contains a $5 bill. The first box contains 3 "lose" envelopes and 2 so-


Problem 4. Let X denote the number of boys in a family with four children. Pr(X > 3) is? a. 5/16 b. ¼ c. 11/16 d. 2/3 e. None of the above

Probability, Statistics & Finance

Problem: The APR for a 30 year, $250,000 mortgage at 6% interest compounded monthly and two discount points is: A. 6.25% b. 6.19% c. 6.12% d. 5.87% e. None of the above.

Probability & Statistics : Chebychev Inequality

Suppose that a probability distribution has a mean 20 and standard deviation 3. The Chebychev inequality states that the probability that an outcome lies between 16 and 24 is: a. less than 7/16 b. at least 7/16 c. at most 1/4 d. at least 1/4 e. none of the above

Probability & Statistics : Outcome of Dice Rolls

Two people play a game. A single die is thrown. If the outcome is a 2 or a 3, then player A pays player B $6.00. How much should B pay A when a 1, 4, 5, or 6 is thrown so that A and B break even, on average, over many repititions of the game? a. $2.00 b. $8.00 c. $4.00 d. $6.00 e. None of the above

Sets, Counting and Probability : Events

A light Bulb manufacturer tests a light bulb by letting it burn until it burns out. The experiment consists of observing how long (in hours) the light bulb burns. Let E be the event "the bulb lasts less than 100 hours", F be the event "the bulb lasts less than 50 hours",and G be the event "the bulb lasts more than 120 hours." T

Selection Without Replacement, Combinations & a Probability Tree

Three letters from A, B, C, D, and E are selected one at a time (without replacement). a. What is the probability that they are selected in alphabetical order? b. What is the probability that they are selected in alphabetical order, if B is the first letter selected.

Frequencies of Dice Roll Combinations and Probabilities of Dice Rolls

When a pair of dice is rolled, the total will range from 2 (1,1) to 12 (6,6). It is a fact that some numbers will occur more frequently than others as the dice are rolled over and over. A. Why will some numbers come up more frequently than others? B. Each die has six sides numbered from 1 to 6. How many possible ways can a nu

Probability : Maximum Likelihood Estimator and p-Values

A GMAT test center reports that out of 200 students who took a GMAT test at the test center 60 scored above 600. a) Derive the maximum likelihood estimator (MLE) of p, the proportion of students scoring above 600. What is the ML estimate of n*p(1-p) b) What is the MLE of P(X<50)? c) Test if the proportion of students sc

How to Set Up a Guessing Problem

Six popular brands of cola are to be used in a blind taste study for consumer recognition. A. If 3 distinct brands are chosen at random from the 6 and if a consumer is not allowed to repeat any answers, what is the probability that all 3 brands could be identified by just guessing? B. If repeats are allowed in the 3 brands

Probability distribution

A single die is rolled once. You lose $12 if a number divisible by 3 turns up. How much should you win if a number not divisible by 3 turns up for the game to be fair? How would I set this up, and what formulas?