Question

The life X (in years) of a regulator of a car has the pdf

32 f(3) = 83 -e-(2/8), 0<x< 0. (a) What is the probability that this regulator will last at least 5 years? (b) Given that it has lasted at least 5 years, what is the conditional probability that it will last at least another 5 years? (c) Suppose the replacement cost Y in dollars) after the regular dies is proportional to X and with mean $5,120. Find the density of Y.

Answer 1

I The life of a regulator of a care hard pdf Ce fcx) z 322 , -() 3 @ 3 The prole that this regulator will last at least 5 years z P(x >5)2 5 322 e - (78) dx = -5/6) =0.483 Das P(X{X) = 1- 2- trs», axo] 6 Now poole that it has lasted 5 years, will survive more 5 years will be given by = P(x>1010x>5) F = P(x>10; *>5), PCX >10) P(x>5) P(x>5) 20.181

( let y denote the replacement cost. Giren y & K3 and Ely)2 5.120

Y = K x 3 csay) dy , 3x²k 2) , 3x'k 2 1J1 2 lgl 2 - 3k x2 3 k. Now fey) s e "? Hoste-es? 1yxo Co. Given ELY) 5:12 2 kg 3 k. 512 2) KI 100 Pof off becomes f(y) t e 7 15:12 5-12