Solving the first posted question according to the HOMEWORKLIB RULES.
That's Q3.
3. Given the survival function: S(t) exp(-t7) derive the probability density function and the hazard function...
Given the survival function: S(t) exp(-£7) Derive X using the definition of the hazard function and basie definition of conditional probability
Given the survival function: S(t)-exp-t)' Derive S(t) using the definition of the hazard function.
Prove that if T has an arbitrary continuous distribution, the cumulative hazard of T, A(T), has an exponential distribution with unit parameter.
Recall that X ∼ Exp(λ) if the probability density function of X is fX(x) = λe−λx for x ≥ 0. Let X1, . . . , Xn ∼ Exp(λ), where λ is an unknown parameter. Exponential random variables are often used to model the time between rare events, in which case λ is interpreted as the average number of events occurring per unit of time. Recall that X ~ Exp(A) if the probability density function of X is fx(x)-Ae-Az for...
5. Let f(t) be the probability density function, and F(t) be the corresponding cumulative f(t) distribution function. Define the hazard function h(t) Show that if X is an 1-F(t): exponential random variable with parameter 1 > 0, then its hazard function will be a constant h(t) = 1 for all t > 0. Think of how this relates to the memorylessness property of exponential random variables.
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...
please show thorough work Problem 3 Recall that X ~ Exp(A) if the probability density function of X is fX(x)-Ae-λ r for z 2 0, Let Xi, , Xn ~ Exp(A), where λ is an unknown parameter. Exponential random variables are often used to model the time between rare events, in which case λ is interpreted as the average number of events occurring per unit of time a) Let zı, . . . , en be n observations of an...
QUESTION6 (a) The three-parameter gamma distribution has the probability density function x (r)- exp (r-c)-1,x> Derive the mean of the distribution. (b) Ifx beta I (m, n) show thatY ax +b(l-X) has the four-parameter beta distribution with parameters a, b, m and n.
Suppose that a given individual in a population has a survival time which is exponential with a hazard rate 0. Each individual's hazard rate θ s potentially different and is sampled from a gamma distribution with density function TCB) Let X be the life length of a randomly chosen member of this popula- tion. (a) Find the survival function of X. (Hint: Find S(x) Ele" .) (b) Find the hazard rate of X. What is the shape of the hazard...
PHYS1047 a) Given a random variable x, with a continuous probability distribution function fx) 4 marks b) The life expectancy (in days) of a mechanical system has a probability density write down equations for the cumulative distribution C(x) and the survival distribution Px). State a relationship between them. function f(x)=1/x, for x21, and f(x)=0 for x <1. i Find the probability that the system lasts between 0 and I day.2 marks i) Find the probability that the system lasts between...