4. Let T = min(Xi, X2,Xy), where each Xi~Weibull(8,a) and X1, X2, X3 are independent (component...
4. Let T = min(Xi, X2,Xy), where each Xi~Weibull(8,a) and X1, X2, X3 are independent (component lifetimes). Show that T also has a Weibull distribution exactly with some shape and scale parameters. (This is so-called "weakest-link model" Hint: find the reliability function of T.)
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3). 3. (25 pts.) Let X1,...
Let T,Tn be independent random variables with Weibull distributions with scale parameters ρι, . . . ,Pn and common shape γ. Prove that T min (T, . . . ,Tn) also has a Weibull distribution with shape y. Derive the distribution of T- min(Ti,...,Tn). man11
11. Let X1, X2, X3 and X4 be independent lifetimes of memory chips. Suppose that Xi N(300, 102) for i = 1, 2, 3, 4 where the parameters are measured in hours. Compute the prob- ability that at least two of the four chips lasts at least 310 hours. (You may leave your answer in terms of an integral, in terms of, or you may leave your answer as an actual real number).
Let X1, X2, X3 … be independent random variable with P(Xi = 1) = p = 1-P(Xi=0), i ≥ 1. Define: N1 = min {n: X1+…+ Xn =5}, N2 = 3 if X1 = 0, 5 if X1 = 1. N3 = 3 if X4 = 0, 2 if X4 = 1. Which of the Ni are stopping times for the sequence X1, …?
14. Let X1, X2, X3 be independent random variables that represent lifetimes (in hours) of three key components of a device. Suppose their respective distributions are exponential with means 1000, 1500, and 2000. Let Y be the minimum of Xi, X2, X3 and compute P(Y 1000).
Let X1,X2 be two independent exponential random variables with λ=1, compute the P(X1+X2<t) using the joint density function. And let Z be gamma random variable with parameters (2,1). Compute the probability that P(Z < t). And what you can find by comparing P(X1+X2<t) and P(Z < t)? And compare P(X1+X2+X3<t) Xi iid (independent and identically distributed) ~Exp(1) and P(Z < t) Z~Gamma(3,1) (You don’t have to compute) (Hint: You can use the fact that Γ(2)=1, Γ(3)=2) Problem 2[10 points] Let...
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
3. Let X1, X2, ..., X, be a random sample from the Weibull distribution with parameters B, 7> O and - < a < oo as shown in your table of distributions. Find the distribution for X (1) = min{X1, X2, ...,Xn}, the minimum value of the sample. (Name it!) (Hint: For help with finding the cdf, see Problem 2 on HW 1.)
Let X1, X2, and X3 be uncorrelated random variables, each with 4. (10 points) Let Xi, X2, and X3 be uncorrelated random variables, each with mean u and variance o2. Find, in terms of u and o2 a) Cov(X+ 2X2, X7t 3X;) b) Cov(Xrt X2, Xi- X2)