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Let T,Tn be independent random variables with Weibull distributions with scale parameters ρι, . . ....
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.)
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. 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 Xo and Xı be independent exponentially distributed random variables with re- spective parameters Ao and ^i, so that, P(Xi t)eAit, for t2 0, i = 0,1 Let 0 if Xo X1, N = 1 if X1X0, min{Xo, X1}, M = 1 - N, V = x{X0, X1}, and W = V -U = |X0 - X1]. and U max Verify that U XN and V XM, then find the following: (a) P(N 0, U > t), for t 2...
1. Let X and Y be two independent exponential random variables with parameters λ and μ, respectively. Compute the probability P(X Y| min(X,Y)-x).
Let the independent normal random variables Y1,Y2, . . . ,Yn have the respective distributions N(μ, γ 2x2i ), i = 1, 2, . . . , n, where x1, x2, . . . , xn are known but not all the same and no one of which is equal to zero. Find the maximum likelihood estimators for μ and γ 2.
et 11, . .. , /n be independent continuous nonnegative random variables with hazard functions λι ( .). . . . , λη (. ). Prove that T-man (Tİ , . . . , Tn) has hazard function Σηι λίο.
3.9. Problem*. (Section 9.1) The following problems concern maximums and minimums of collections of independent random variables. (a) Let Y.Y2, ..., Yn be independent exponential random variables with parameters 11, 12,..., In, respectively. Prove that E[min{Yı, Y2, ..., Yn}] < min{E[Y], E[Y2),..., E|Y.]} (b) Suppose that X1, X2, ..., X, are independent continuous random variables with uni- form distributions on (0,1). Compute E[min{X1, X2, ..., Xn}] and E[max{X1, X2,..., X.}]
4a). Let X1 and X2 be independent random variables with a common cumulative distribution function (i.e., c.d.f.) F(y) = { 0" if0cyotherwise。 Find the p.d. f. of X(2,-max(X, , xa). Are X(1)/X(2) and X(2) independent, where X(1,-min(X,, X2) ? 4a). Let X1 and X2 be independent random variables with a common cumulative distribution function (i.e., c.d.f.) F(y) = { 0" if0cyotherwise。 Find the p.d. f. of X(2,-max(X, , xa). Are X(1)/X(2) and X(2) independent, where X(1,-min(X,, X2) ?
A amogos lf X and Y are independent exponential random variables with parameters 11 and 12 respectively, compute the distribution of Z = min(X,Y). What is the conditional distribution of Z given that Z = X?