Question

Thetime(inhours)requiredtorepairamachineis an exponentially distributed random variable with parameter λ = 1 . What is 2 (a) the probability that a repair time exceeds 2 hours? (b) the conditional probability that a repair takes at least 10 hours, given that its duration exceeds 9 hours?

Answer 1

a) 0.1353

b) 0.3679

Explanation:

Let's start by defining the random variable T.

T : ''The time (in hours) required to repair a machine''

T ~ exp (λ)

T ~ exp (1)

The probability density function for the exponential distribution is

(In the equation I replaced λ = L)

`f(x)=Le^(-Lx)`

With L > 0 and x ≥ 0

In this exercise λ = 1 ⇒

`f(x)=e^(-x)`

For a)

`P(T>2)`

`P(T>2)=1-P(T\leq 2)`

`P(T>2)=1-\int\limits^2_0 {e^(-x) } \, dx`

`P(T>2)=1-(-e^(-2)+1)`

`P(T>2)=e^(-2)=0.1353`

For b)

`P(T\geq 10/T>9)`

The event (T ≥ 10 / T > 9) is equivalent to the event T ≥ 1 so they have the same probability of occur

`P(T\geq 10/T>9)=P(T\geq 1)`

`P(T\geq 1)=1-P(T<1)=1-\int\limits^1_0 {e^(-x) } \, dx`

`P(T\geq 1)=1-(-e^(-1)+1)=e^(-1)=0.3679`