Question

2. Light bulbs are known to have an average lifetime of 2,000 hours. Suppose we model the lifetime of a light bulb by the following probability density function with (yet unknown) parameter c: p(t) = 1-e-t/c when t20 and p(t) = 0 otherwise. (a) Determine the value of the parameter c so that the probability density function has mean 2,000 hours. (b) Determine the probability a lightbulb fails before 1,500 hours. (C) Suppose the lightbulb has already been on for 1,000 hours, what is the probability that the lifetime of the lightbulb is less than 2,000 hours.

Answer 1

There is a small hypo in question, the given plt) is probability distoibution function Bobability density function say fit, obtained by differentiating probability distribution function pit). that is f(t): 1 o , otherwise Sanden rasiable by X. Let us denote underlying Sandom (a) Mean of distribution o zle letle +30 E(X) = på foda = L xétleda se → C = 2000 hos. b) Requised probability Pr (X $ 1500 hos) - N PC (1500 hrs) 1500 hrs) =- 34 = 0.5276 (C) Required probability Po ( x < 2000hrs / x2 1000hrs) Pr ( 1000 <x < 2000) Usicas Bayes theoxim Pr ( X 1000 hes) Pr ( x < 2000) - (x <1000) Pr (* 1000) p(2000) - Þ(1000) 1- Pret <10007 Þ(2000) – Þ(1000) 1- € (1000) obli-éla) 1-0) ola 0.6065 2- € =