Let X be the number of material anomalies occurring in a particular region of an aircraft gas-turbine disk. The article "Methodology for Probabilistic Life Prediction of Multiple-Anomaly Materials"+ proposes a Poissondistribution for X. Suppose that μ = 4. (Round your answers to three decimal places.)(a) Compute both P(X ≤ 4) and P(X < 4).P(X ≤ 4) =P(X < 4) =(b) Compute P(4 ≤ x ≤ 9).(c) Compute P(9 ≤ X).(d) What is the probability that the number of anomalies does not exceed the mean value by more than one standard deviation?

Let X represents the number of material anomalies occurring in a particular region of an aircraft…

Answered: Let X be the number of material…

A First Course in Probability (10th Edition)

10th Edition

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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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Once an individual has been infected with a certain disease, let X represent the time (days) that elapses before the individual becomes infectious. An article proposes a Weibull distribution with a = 2.6, B = 1.8, and y = 0.5. [Hint: The two-parameter Weibull distribution can be generalized by introducing a third parameter y, called a threshold or location parameter: replace x in the equation below, a -x« – le-(x/B)ª Ba x 2 0 f(x; а, B) %3D x y.] (a) Calculate P(1 1.5). (Round your answer to four decimal places.) 0.8050 (c) What is the 90th percentile of the distribution? (Round your answer to three decimal places.) 2.980 |× days (d) What are the mean and standard deviation of X? (Round your answers to three decimal places.) 1.594 х days mean standard deviation 0.585 X days
Let X; € {1,2, 3, ...} be the number of days until relapse for patient i who is diagnosed with multiple sclerosis and currently in remission. We model this data using a geometric distribution with pmf iid fx\p(x |p) = (1 – p)*-'p x-1 X1, X2, for 0
Let X1, , Xµ be iid with population density (1 0) I>0, Sx(x) = %3D otherwise. Here 0 is an unkown population parameter. 0 has an Exponential(1) distribution. Find the method of moment estimator for 0. Let's call this 6. Is ô unbiased for 0 ? Explain with precise computation. Show that X Find the maximum likelihood estimator for 0. Let's call this 62. Is ô2 unbiascd for 0 ? Explain with precise computation.
Can someone please explain it to me ASAP??!!!
Q 4.4. An individual picked at random from a population has a propensity to have accidents that is modelled by a random variable Y having the gamma distribution with shape parameter a and rate parameter ß. Given Y = y, the number of accidents that the individual suffers in years 1, 2, ..., n are independent random variables X₁, X2,... Xn each having the Poisson distribution with parameter y. (a) Write down a function f so that the joint distribution of Y, X₁, Xn can be described via P(a ≤ Y ≤ b, X₁ = k1, X₂ = k2 … . . Xn = kn) = [° ƒ (y, k1, k2, ... kn)dy and derive from this expression that, for your choice of f, Y has the Gamma distribution, and that conditionally on Y = y, X₁, X2,... Xn are independent, each having the Poisson distribution with parameter y.
A1 We mentioned the use of the Poisson distribution for modelling the number of “events” that occur in a two-dimensional region. Assume that when the region R being sampled has area a(R), the number X of plants of a certain species occurring in R has a Poisson distribution with parameter λa(R) (where λ is the expected number of plants per unit area) and that nonoverlapping regions yield independent X’s. Suppose an ecologist selects n nonoverlapping regions R1, . . ., Rnand counts the number of plants of a that species found in each region. Which is the expression of the joint pmf (likelihood)? Which is the maximum likelihood estimator of λ.
Let X; € {1,2, 3, ...} be the number of days until relapse for patient i who is diagnosed with multiple sclerosis and currently in remission. We model this data using a geometric distribution with pmf iid X1, X2,..., Xn fxp(x | p) = (1 – p)*-'p for 0 < p < 1 defined on x E {1,2, 3, ...} and 0 elsewhere. Here, p is the risk of relapse on each day. 1. Using a p~ Beta(a, B) prior, derive the posterior density of p, fp|X, (p | Xn).
Let 0 be the maximum likelihood estimator of the parameter 0. We are given that lim, 00 E(0) = 0 and 1. Then what is the asymptotic distribution of 0? %3D CRLB(@) Var(0) lim 00 O a. @~N(0,CRLB(0)). O b. ê- N(0, CRLB@) Oc O~ N(0,CRLB(0)). O d. - N(ô. CRLB(@).
7.
6. Find the median of an exponential random variable X with parameter 1 = 4. (The median of X is m iff P(X < m) = 0.5.) Does it differ from E(X)? Try to explain.
page 328 5.2.2

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