Question

The lifetime of an electronic component in an HDTV is a random variable that can be modeled by the exponential distribution with a mean lifetime ß. Two components, X1 and X2, are randomly chosen and operated until failure. At that point, the lifetime of each component is observed. The mean lifetime of these two components is X1 + X2 X =- a) Find the probability density function of x using the MGF technique (the method of moment-generating functions). b) If the mean lifetime of the electronic component is two years (B = 2), what is the probability that the mean lifetime of two tested components will be more than three years? You may use R and/or additional software to calculate your answer to this question, but please show the relevant equation(s) used to obtain the final answer.

Plz use MGF technique

Answer 1

Solution :- from the ginen data, 3 +*, - 2- 1982 – Exp (18) Mx, (t) = Meg(t) = T-BE =) By uring MGF method, we get > Mã (t) = Eceti) - € (et (2013) - € Celle (21+x3) | = (a + 1) x ( 49 ). > Mact) - € (21) x € (2522) > ma (1)= ()*(*)-) 5 ~ Gamma ( 2) b if B= 2, then :.*~ Gamma (12) - The probability that the mean lifetime of the two components tested will be more than 3 years P(123) - | - P(133) = 1 -0.800852 -10.199148)