Concept explainers
Let X denote the lifetime of a component, with f (x) and F(x) the
An increasing failure rate function indicates that older components are increasingly likely to wear out, whereas a decreasing failure rate is evidence of increasing reliability with age. In practice, a “bathtub-shaped” failure is often assumed.
a. If X is exponentially distributed, what is r(x)?
b. If X has a Weibull distribution with parameters α and β, what is r(x)? For what parameter values will r(x)be increasing? For what parameter values will r(x) decrease with x?
c. Since
so that if a component lasts β hours, it will last forever (while seemingly unreasonable, this model can be used to study just “initial wearout”). What are the cdf and pdf of X?
Trending nowThis is a popular solution!
Chapter 4 Solutions
Probability and Statistics for Engineering and the Sciences
- LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.arrow_forwardLet the pdf of X be f(x)=3x^2, 0<x<1. Find the probability that X is greater than 0.3.arrow_forwardLetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise (a) Show that the moment generating function mX(s) :=E(esX) =λ/(λ−s) for s< λ;arrow_forward
- Let the pdf of X be defined by f(x) = ke−0.3x for all positive values of x and for some constant k. Solve for the following: a) mean of X b) variance of Xarrow_forwardLet X1 ... Xn i.i.d random variables with Xi ~ U(0,1). Find the pdf of Q = X1, X2, ... ,Xn. Note that first that -log(Xi) follows exponential distribuition.arrow_forwardLet x~ possion(alpha). Show that E[x(x-1)(x-2)....(x-k)]=ak+1arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage