Question

3. The time in days between breakdowns of a machine is exponentially distributed with λ-02 a. What is the expected time between machine breakdowns? b. What is the probability that after the machine is repaired, it lasts at least five days before failing again? C. If the machine has performed satisfactorily for seven days, what is the probability that it lasts nine days before breaking down?

Answer 1

Let us denote the breakdown of a Machine by X.

Then X~E(0.2).

a) Expected time between Machine Breakdown= E(X)= 1/0.2=5 days. (Since Mean of the exponential distribution is just reciprocal of its parameter.)

b) P(X>=5)= 1-P(X<=5) = 1- 0.6321206 = 0.3678794

c) P(X<=9| X>=7)= P(7<X<9) = P(X<9)- P(X<7)= 0.08129808

Here, I calculated the above probability by using R. So, I am attaching my R-Code:

> pexp(5,0.2)
[1] 0.6321206
> 1-pexp(5,0.2)
[1] 0.3678794
> pexp(9,0.2)
[1] 0.8347011
> pexp(7,0.2)
[1] 0.753403
> pexp(9,0.2)-pexp(7,0.2)
[1] 0.08129808