Question

Let X represent the amount of time until the next student will arrive in the library parking lot at the university. The distribution of arrival time can be modeled using an exponential distribution with rate 1/4 arrivals per minute. Find the probability that it will take more than 10 minutes for the next student to arrive at the library parking lot. Group of answer choices 0.082085 0.329680 0.917915 0.670320

Answer 1

Answer:

Probability that it will take more than 10 minutes for the next student to arrive at the library parking lot is 0.082085.

Step-by-step explanation:

We are given that the distribution of arrival time can be modeled using an exponential distribution with rate 1/4 arrivals per minute.

Let X = Amount of time until the next student will arrive in the library parking lot at the university.

The probability distribution for exponential distribution is given by;

                        $f(x) = \lambda e^{-\lambda x} ; x >0$

where, $\lambda$ = parameter of this distribution or the arrival rate = $\frac{1}{4}$ = 0.25

So, X ~ Exp($\lambda = 0.25$)

Now, to find the less than or greater than probabilities in exponential distribution we use the Cumulative distribution function of exponential function, i.e.;

                     $F(x) = P(X \leq x) = 1 - e^{-\lambda x} ; x >0$

            

So, probability that it will take more than 10 minutes for the next student to arrive at the library parking lot is given by = P(X > 10 minutes)

     P(X > 10) = 1 - P(X $\leq$ 10)

                     = 1 - [ $1 - e^{-0.25 \times 10}$ ]

                      = $e^{-2.5}$ = 0.082085

Therefore, probability that it will take more than 10 minutes for the next student to arrive is 0.082085.