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 Poisson distribution for X. Suppose that ? = 4. (Round your answers to three decimal places.) (a) Compute both P(X ≤ 4) and P(X < 4). (b) Compute P(4 ≤ X ≤ 7). (c) Compute P(7 ≤ X). (d) What is the probability that the number of anomalies does not exceed the mean value by more than one standard deviation?

Question 3. Let X₁,, Xn be a random sample from a distribution with the pdf given by f₁₁x(x) = exp(-ª) if x ≥ A, otherwise f(x) = 0, where > 0. Find the MLE's of 0 and X. Start by writing the likelihood function and note the constraint involving A. Question 4. #4.2.9 of the textbook.

A random variable X is distributed as Poisson distribution with λ = 3.5. Usethis Poisson distribution (Table A.3) to determine the following probabilities: (a)P(X < 5), (b) P(2 ≤ X ≤ 6), (c) P(X > 7), (d) P(X ≥ 5).

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.

23. Find the cumulant generating function of gamma distribution and obtain various cumulants.

When taking random samples of n observations from a population that is not normally distributed, the sampling distribution, x, will be approximately normally distributed never. The population itself must be normally distributed for x to be normally distributed. when the population distribution has low kurtosis. always. when the population standard deviation, o, is greater than n. when n is sufficiently large.

Suppose x has a distribution with u = 29 and o = 25. (a) If a random sample of size n = 43 is drawn, find u, o , and P(29 < x < 31). (Round o, to two decimal places and the probability to four decimal places.) 0こ= P(29 < x < 31) = (b) If a random sample of size n = 61 is drawn, find u, o y and P(29 < x < 31). (Round o , to two decimal places and the probability to four decimal places.) 長= 『ス= P(29 < x < 31) =

If a project has an expected time of completion Te of 45 weeks witha standard deviation of 7 weeks, what is the probability of completing it (a) within one year (52 weeks)? (b) within 38 weeks.