Question

The time between arrivals of taxis at a busy intersection is exponentially distributed with a mean of 24 minutes. (a) What is the probability that you wait longer than one hour for a taxi? (b) Suppose you have already been waiting for one hour for a taxi, what is the probability that one arrives within the next 10 minutes? Round your answers to 4 decimal places.

Answer 1

Answer:

$P(X>60) = 0.0821$

$P(X<70 |X>60) = 0.3408$

Step-by-step explanation:

Given that:

X ~ Exp (β=24)

What is the probability that you wait longer than one hour for a taxi?

Pdf: f(x) = $\frac{1}{\beta }e^{(-x/ \beta) }$   , 0 < x

Cdf: P(X ≤ x) =  $1-e^{(-x/ \beta)}$

P(X  ≤ x) =$1-e^{(-x/ 24)}$         for x > 0 ⇒ P ( X > x) = $e^{-x/24}$

$P(X>60) = exp (-60/ \beta)\\P(X>60) = e^{-25}\\P(X>60) = 0.082085$

$P(X>60) = 0.0821$

Thus, the probability that you wait longer than one hour for a taxi = 0.0821

b)

$P(X<70 |X>60)= P(X<70-60)\\P(X<10) =1-exp(-10/ \beta)\\P(X<10) = 1-e^{-0.416667$

$P(X<10)$ ≅ 0.340759

$P(X<10)$  = 0.3408

$P(X<70 |X>60) = 0.3408$

The probability that one arrives within the next 10 minutes = 0.3408