(1 point) Suppose that the life distribution of an item has hazard rate function λ(t)-332, t...
Suppose that the life distribution of an item has the hazard rate function (t) = t3, for t > 0. What is the probability that a 1 year old item will survive to age 2?
Suppose X has a Poisson(λ) distribution (a) Show that E(X(X-1)(X-2) . .. (X-k + 1)} for k > 1. b) Using the previous part, find EX (c) Determine the expected value of the random variable Y 1/(1 + X). (d) Determine the probability that X is even. Note: Simplify the answers. The final results should be expressed in terms of λ and elementary operations (+- x ), with the only elementary function used being the exponential
3. Classifying Life Distributions. Suppose a continuous lifetime T has survival function S(O), hazard function h(i), cumulative hazard function (1), and mean residual life m(t). Consider the following properties that I might have: I. h(t) is nondecreasing for 120, called increasing failure rate (IFR). II. HIV/1 is nondefreasing for >0, called increasing failure rate on the average (IFRA). II. ml) Sm(0) for all / 20, called new better than used (NBU). IV. m(1) decreases in 1, called decreasing mean residual...
1. X,,x2,..., X, is a random sample from a Poisson (0) distribution with probability mass function 0*e f(x) = x=0,1,..., 0 >0. x! (1) Write Poisson (0) as an exponential family of the form fo(x) = exp{c(0)T(x)-v (0)}h(x) State what c(0), 7(x), and y (@) are. (ii) a. Prove that for the exponential family given in (i), E[T(X)]=y'(c). b. Hence find the mean of the Poisson (0) distribution. [3] [6] [2] 21 (iii) Show that for the Poisson (0) distribution,...
li iNA/113 リ1에 1.st).. T,li(t)ls)11(aliini TT Previous Problem List Next (1 point Consider the function f(z)-21 +3. (a) Give the piecewise linear function that corresponds to the absolute value function. iii , then f(z)- f z > l then f(r)- (b) Choose the correct corresponding graph. f(n) fin) B. 3-2-1 f(x) C. 3/2-1 f(x) D.
3. An infinite bar has initial temperature distribution: T(x,0)-T0 [s(x-l) +δ(x + 1)] Find T(x,t) for >0. The free space Green's function for the 1-D diffusion equation is G(x -x)- e 4DI 4TDt
1. Consider the following Cumulative Distribution Function (CDF) of random variable 0.41 1 t <3 0.78 3 < t < 5 0.94 5t<7 F(t) = a. 4 Find P(T> 3); P(1.5 < T b. [3] Find E(3T +5) and V (3T5) 6); P(T < 5IT2)
(1 point) The random variable T has the following CDF: O for t <3 F(t) = 0.1 for 3 <t<6 0.8 for 6 <t<10 | 1 for t > 10. Please answer to 3 decimal places. Part a) Compute E(T) Part b) Compute Var(7)
1. The Weibull distribution has many applications in reliability engineering, survival analysis, and general insurance. function Let β > 0, δ > O. Consider the probability density fx(x) = βδΧ6-1 e-Px®, x>0, zero otherwise. Find the probability distribution of W-X" b) Determine the probability distribution of W by finding the p.d.f. of W, fw(w), using the change-of-variable technique D) Find the p.d.f. of W, w)-x(w ii What is the name of the probability distribution of W? What are its parameters?...
1. The Weibull distribution has many applications in reliability engineering, survival analysis, and general insurance. function Let p> 0, δ > 0. Consider the probability density x>0 zero otherwise. Find the probability distribution of w-x6 a) Determine the probability distribution of W by finding the c.d.f. of W, Fw(w). Find the cd.f. of X, Fx(x) = P(X x). “Hint', 1: u-substitution: u "Hint" 2: There is no such thing as a negative cumulative distribution function "Hint" 3: Should be Fx(0)-0,...