Question

s. Suppose the random variable represents the time to failure (in thousands of miles driven) of the signal lights on an automobile, and that has a Weibull distribution with and a. What is the probability that the signal lights function for at least 20,000 miles? b. For how many miles can the signal lights be expected to last? Part a can be answered with or without R. Answer part b without using R.

Suppose the random variable X represents the time to failure (in thousands of miles driven) of the signal lights on an automobile, and that X has a Weibull distribution with alpha = .0125 and Beta = 2/3.

A) What is the probability that the signal lights function for at least 20,000 miles?

B) For how many miles can the signal lights be expected to last?

Answer 1

A)

X ~ Weibull($\alpha$ = 0.0125, $\beta$ = 2/3)

Probability that signal lights function for at least 20,000 miles = P(X $\ge$ 20,000)

= exp[-(20000 / $\beta$)^alpha]

= exp[-(20000 / (2/3))^0.0125]

= 0.320609

B)

Expected time to failure (in thousand of miles) = E(X) = $\beta \Gamma \left ( 1 + 1 / \alpha \right )$

$= (2/3) \Gamma \left ( 1 + 1 / 0.0125 \right )$

$= (2/3) \Gamma \left ( 1 + 80 \right )$

$= (2/3) \Gamma \left ( 81 \right )$