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kness of a chip (X) has a mean of 10 micrometers and chev's inequality, find a bound for the probability that 14 micrometers. t for some a and b, P(a < X < b) is at least 8/9. Find

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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The following table shows the returns of two types of investments, stocks (x) and bonds (y), under two different scenarios: boom and recession. The frequency f(∙) indicated the probability of boom and recession, and it is the same for any type of investment What is the expected return from bonds? What is the risk (standard deviation) of a bonds investment?
The mean number of patients admitted per day to the emergency room of a small hospital is 0.5. If, on any given day, there are only 6 beds available for new patients, what is the probability that the hospital will not have enough beds to accommodate its newly admitted patients? P(Y > 1-) (formula) (numeric result) Use: y for the value of the variable )/ ) for fractions (O! for factorial ) for the exponential function e^(
'Time headway' in traffic flow is the elapsed time between the time that one car finishes passing a fixed point and the instant that the next car begins to pass that point. Let X be the time headway for two randomly chosen consecutive cars (in seconds). Suppose X has pdf f(x) = {* F(x) (a) Find the value of k. (b) Obtain the mean value of headway and the standard deviation of headway. (c) The cdf of X is = x > 1 x ≤ 1 0 1- x ≤ 1 x>1 Using the cdf, determine the following probabilities. (i) What is the probability that observed depth is at most 3? (ii) What is the probability that observed depth is between 3 and 2?
At the end of summer, the total weight of seeds accumulated by a nest of seed-gathering ants will vary from nest to nest. If the total weight of seeds accumulated by a nest is exponentially distributed with parameter λ = 1/5, (a) What is the probability that the total combined weight of the seeds gathered by 100 nests will be larger than 4 95 pounds by the end of next summer? (b) What is the probability that the average weight of the seeds gathered by the 100 nests is larger than 5.3 ? (c) What assumptions are you making to answer parts (a) and (b)? Do you think those assumptions make sense in the context of this problem? Explain. (d) Tell us about the possibility that the total combined weight of the seeds gathered by 2 nests follows a normal distribution. Justify your answer in the speciﬁc context of this problem (ants, nests, seed-gathering). That is, do not just make generic statements that you think apply to every context in the world.
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The "kernel trick" is a quick way to integrate when you can recognize a distribution in some equation g(x) which you want to integrate. It takes advantage of the fact that the pdf f(x) of a proper probability distribution must integrate to 1 over the support. Thus if we can manipulate an equation g(x) into the pdf of a known distribution and some multiplying constant c in other words, g(x) = c · f(x) --- then we know that C • --- Saex 9(x)dx = c Smex f(x)dx = c ·1 = c Steps: 1. manipulate equation so that you can recognize the kernel of a distribution (the terms involving x); 2. use the distribution to figure out the normalizing constant; 3. multiply and divide by the distribution's normalizing constant, and rearrange so that inside the integral is the pdf of the new distribution, and outside are the constant terms 4. integrate over the support. The following questions will be much easier if you use the kernel trick, so this question is intended to give you basic practice. QUESTION:…
The mean number of patients admitted per day to the emergency room of a small hospital is 2.5. If, on any given day, there are only 3 beds available for new patients, what is the probability that the hospital will not have enough beds to accommodate its newly admitted patients? P(Y > 1 (formula) (numeric result) Use: y for the value of the variable ) for fractions ( )! for factorial ) for the exponential function e^( Note: You can earn partial credit on this problem.
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(16) Let X U(3) find Var(-x) and distribution function.
Please solve (a) & (b) only!
44.The time (in hours) required to repair a machine is exponentially distributed with parameter = what is the probability that a repair takes at least 10 hours 2 given that its duration exceeds 9 hours?