Question

Let X be the waiting time for a car to pass by on a country road, where X has an average value of 35 minutes. If the random variable X is known to be exponentially distributed, what is the probability that the wait time is greater than 37 minutes

Answer 1

Probability that the wait time is greater than 37 minutes is 0.3474.

Explanation:

We are given that the random variable X is known to be exponentially distributed and X be the waiting time for a car to pass by on a country road, where X has an average value of 35 minutes.

Let X = waiting time for a car to pass by on a country road

The probability distribution function of exponential distribution is given by;

`f(x) = \lambda e^(-\lambda x) , x >0` where,
`\lambda` = parameter of distribution.

Now, the mean of exponential distribution is =
`(1)/(\lambda)` which is given to us as 35 minutes that means
`\lambda = (1)/(35)` .

So, X ~ Exp(
`\lambda = (1)/(35)` )

Also, we know that Cumulative distribution function (CDF) of Exponential distribution is given as;

`F(x) = P(X \leq x) = 1 - e^(-\lambda x)` , x > 0

Now, Probability that the wait time is greater than 37 minutes is given by = P(X > 37 min) = 1 - P(X
`\leq` 37 min)

P(X
`\leq` 37 min) =
`1 - e^{-(1)/(35) * 37}` {Using CDF}

= 1 - 0.3474 = 0.6525

So, P(X > 37 min) = 1 - 0.6525 = 0.3474

Therefore, probability that the wait time is greater than 37 minutes is 0.3474.