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

Answer:

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

Step-by-step 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 = $\frac{1}{\lambda}$  which is given to us as 35 minutes that means  $\lambda = \frac{1}{35}$  .

So, X ~ Exp( $\lambda = \frac{1}{35}$ )

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) = $1 - e^{-\frac{1}{35} \times 37}$        {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.