The lifetime of a certain electronic device is a random variable X (years) and has a probability density function given by f(x) = (1/a) * exp(-x/a), x ≥ 0, where a > 0 is regarded as a measure of quality.
a. Determine the distribution function of X. Enter an expression in t. [1 mark]
F(t) =
b. Find the expectation of X. Enter an expression in a. [1 mark]
E(X) =
c. Find the variance of X. Enter an expression in a. [1 mark]
Var(X) =
d. Each device is accompanied by a guarantee that it will be replaced if it fails within time t0. For devices of quality a = 2, what value of t0 (in years) limits the manufacturer's liability to the replacement of 15 percent of tubes? Enter a number correct to two decimal places. [2 marks]
t0 =
e. What measure of quality a should the manufacturer strive to obtain if they wish to give a guarantee for 6 months while still restricting their liability to the replacement of 15 percent of tubes? Enter a number to two decimal places. [1 mark]
a =