Chapters: Normal and Exponential random variables.Q: Suppose the time to failure (in years) for a particularcomponent is distributed as an exponential randomvariable with rate 1 = 1/5. For better performance, thesystem has two components installed, and the systemwill Iwork as long as either components installed, andthe system will work as long as either component isfunctional. Assume the time to failure for the twocomponents is independent. What is the probabilitythat the system will fail before 10 years has passed?

Answered: Q: Suppose the time to failure (in…

A First Course in Probability (10th Edition)

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Author:Sheldon Ross

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Chapter1: Combinatorial Analysis

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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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A random selection of volunteers at a research institute have been exposed to a typical cold virus. After they started to have cold symptoms, 10 of them were given multivitamin tablets formulated to fight cold symptoms. The remaining 10 volunteers were given placebo tablets. For each individual, the length of time taken to recover from the cold is recorded. At the end of the experiment the following data are obtained. Days to recover from a cold Treated with multivitamin 2.6, 5.9, 5.3, 5.8, 8.7, 6.2, 7.6, 6.4, 8.0, 4.0 Treated with placebo 3.0, 5.1, 4.7, 4.2, 4.7, 5.6, 5.9, 4.7, 7.1, 4.9 It is known that the population standard deviation of recovery time from a cold is 1.8 days when treated with multivitamin tablets, and the population standard deviation of recovery time from a cold is 1.5 days when treated with placebo tablets. It is also known that both populations are approximately normally distributed. The researchers claim that the mean recovery time, l, of the patients treated…
In a study in pre-term infants, a group where breast feeding is difficult because the mother gets home before the baby does, you run a logistic regression model where the independent variable is the mother's age and the dependent variable is whether the infant was breast feeding at discharge from the hospital. If the log odds for breast feeding for a 20 year old mom was -3.654 + (20) * 0.157 = -0.514, compute the predicted probability for breast feeding for a 20 year old mom. 0.598 / (1+0.598) O 0.598 0.598 / (1 - 0.598) -0.514 / (1+0.598)
Consider a system with one component that is subject to failure, and suppose that we have 120 copies of the component. Suppose further that the lifespan of each copy is an independent exponential random variable with mean 30 days, and that we replace the component with a new copy immediately when it fails. (a) Approximate the probability that the system is still working after 4500 days.Probability ≈ (b) Now, suppose that the time to replace the component is a random variable that is uniformly distributed over (0,0.5). Approximate the probability that the system is still working after 5400 days.Probability ≈
Calculate the variance of a random variable X whose characteristic function is (1+ e3it)² f(t) = 4
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The number of times that a person contracts a cold in a given year is a Poisson random variable with parameter λ=5. Suppose that a new wonder drug (based on large quantities of vitamin C) has just been marketed that reduces the Poisson parameter to λ=3 for 70 percent of the population. For the other 30 percent of the population, the drug has no appreciable effect on colds. If Peter tried the drug for a year and has 3 colds in that time, what is the probability that the drug is beneficial for him?
In experiment of flipping two coins, each coin has two faces: Head and Tail. Each face is being equally likely to be drawn. If two random variables X and Y are defined as X(s) = Number of Heads appeared on both coins Y(s) = Number of Tails appeared on both coins The value of pxy (2,0) is 00.5 00 01 00.25
Let a be a random variable representing the percentage of protein content for early bloom alfalfa hay. The average percentage protein content of such early bloom alfalfa should be u = 17.2%. A farmer's co-op is thinking of buying a large amount of baled hay but suspects that the hay is from a later summer cutting with lower protein content. A small amount of hay was removed from each bale of a random sample of 50 bales. The average protein content from the samples was determined by a local agricultural college to be -15.8% with a sample standard deviation of S = 5.3%- At a = .05, does this hay have lower average protein content than the early bloom alfalfa?
A random variable has poisson distribution such that P (X = 3) = P (X= 4). Find p (6).
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Pi- P2 Z. = %3D SE Suppose a drug company is developing a vaccine, designed to protect against a particular virus. The company states that the vaccine will be equally effective for men and women. In its initial stage, a researcher selects a random sample of 200 men and 100 women independently. At the end of the study, 15 of the men and 10 of the women showed symptoms of the virus after taking the vaccine. Is there enough evidence to conclude that there is a significant difference in the proportions of men and women who show symptoms of the virus after taking the vaccine? (a) State the appropriate hypotheses to answer this question. (b) Compute the test-statistic. (c) (i) Calculate the p-value. (ii) Interpret the p-value you have found. (d) State your conclusion in context of the situation. Page 1 135
Suppose model (XY, XZ, YZ) holds in a 2 x 2 x 2 table, and the common XY conditional log odds ratio at the two levels of Z is positive If the XY and YZ conditional log odds ratios are both positive or both negative, show that the XY marginal odds ratio is larger than the XY conditional odds ratio.

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