On average, a certain kind of kitchen appliance requires repairs once every four years. Assume that the times between repairs are exponentially distributed. Which of the following Excel commands computes the probability that the appliance will work no more than three years without requiring repairs?
A) EXPON.DIST(3,1.0,1)
B) EXPON.DIST(3,0.25,1)
C) EXPON.DIST(3.1.0,0)
D) EXPON.DIST(3.0.25,0)

Question

Quizplus · Accepted Answer

The probability density function for the exponential distribution is given by f(x) = λe^(-λx), where λ is the rate parameter. The cumulative distribution function (CDF) is obtained by integrating the PDF from 0 to x:
F(x) = ∫[0,x] λe^(-λt) dt = 1 - e^(-λx)
We are given that the appliance requires repairs once every four years, so the rate parameter λ = 1/4. The probability of the appliance working no more than three years without requiring repairs is equivalent to the probability of the time between repairs being less than or equal to three years. Therefore,
P(X ≤ 3) = F(3) = 1 - e^(-λx) = 1 - e^(-0.25*3) ≈ 0.427
The Excel command to compute this probability is EXPON.DIST(3,0.25,1), which corresponds to the exponential distribution with rate parameter λ = 0.25, evaluated at x = 3.

On Average, a Certain Kind of Kitchen Appliance Requires Repairs