Question

Problem No. 6.4 / 10 pes. The lifetime, in years of a certain type of pump is a random variable with probability density function .x20 0 True (Note: "True" means "Otherwise" or "Elsewere") 1) What is the probability that the pump lasts more than 3 years? 2) What is the probability that the pump lasts between 1 and 2 years? 3) Find the mean lifetime. 4) Find the variance of the lifetime. 5) Find the cumulative distribution function of the lifetime. 6) Find the median lifetime. 7) Find the 70.-th percentile of the lifetime. Show your explanations Displaying only the final answer is not enough to get credit. Note: round calculated numerical values to the fourth decimal place where applicable. Solution (Show all intermediate steps, formulas, calculations, explanations and comments below this line. Don't write above this line)

Answer 1

3 3 Plosx 53) - .dx. (ltaju let taat da=dt 3.dt. th 12tcy 1 4 - -4+1 t -47 = (1) -»[61522) POS 3) 0.98 1-0.98 probability last more than 3 than 3 years = 0.02 16 prisxea). JAM b 3.dy (1+234 3 3 dt 3 ta 3 2 y --(03-2) 0.08 © Mean Life time - 1 x fxCw).de. 154) dx dxo (1+x)" 3n du 2. (cat) (+1) 0 eet taal dx=dt X-t-1 i ༣།། 3(t-1) dan -14 Wy) dt. 1 t CS Scanned with CamScanner

- 3 -3 1 2 Mean time=0.5 2 MSV Variance (Mean) - 3х2 MSV x? &G).dx .dx (1+2+)4 0 x=t-l let 1+x=t 1 Laco : 3 (t-D² da 2.de + 2 t4 =3 "(t at tahi = 2 (10)- (+*+"] 21 variance 0.75 0.25 - 7 t+! 3. da (2+04 - CDF= I er at [H2-6] stul (ul -1 CS Scanned with CamScanner

CDF = -1 Cetuva 1 CDF - 2 (I+t) Life time - Maximum of pdf 6 Medion Maxl M cu ) (1+) Medion Life time - 1 itong da = 0.0 let ita=t. 7 (1+)4 0 t-1 df 0.07 j t 1 tt = 0.07. *100), [el---] 0.21 jo 1-0.21 0.71 11 H-1)? 3 4-1) 1.408 t-1 + 1.211 CS South CamScahner lifetime percentile t= 2.21