Question

The time between calls made to Amazon's customer service is exponentially distributed with a mean of 10 minutes (taken between each two successive calls). The probability that the time between the next two calls is between 3 minutes and 7 minutes is:
A. 0.7558
B. 0.5034
C. 0.2442
D. 0.2592

Answer 1

Answer:

Probability that the time between the next two calls is between 3 minutes and 7 minutes is 0.2442.

Step-by-step explanation:

We are given that the time between calls made to Amazon's customer service is exponentially distributed with a mean of 10 minutes.

Let X = time between calls made to Amazon's customer service

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 10 minutes that means  $\lambda = \frac{1}{10}$ .

So, X ~ Exp( $\lambda = \frac{1}{10}$ )

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 time between the next two calls is between 3 minutes and 7 minutes is given by = P(3 min < X < 7 min)

P(3 min < X < 7 min) = P(X < 7 min) - P(X $\leq$ 3 min)

P(X < 7) = $1 - e^{-\frac{1}{10} \times 7}$ = 1 - 0.4966 = 0.5034    {Using CDF}

P(X $\leq$ 3) = $1 - e^{-\frac{1}{10} \times 3}$ = 1 - 0.7408 = 0.2592

Therefore, P(3 min < X < 7 min) = 0.5034 - 0.2592 = 0.2442.

Hence, probability that the time between the next two calls is between 3 minutes and 7 minutes is 0.2442.

Answer 2

Answer:

The probability that the time between the next two calls is between 3 minutes and 7 minutes is 0.2442.

Step-by-step explanation:

Let X = time between calls made to Amazon's customer service.

The average time between calls is, β = 10 minutes.

The random variable X follows an Exponential distribution with parameter $\lambda=\frac{1}{\beta}=\frac{1}{10}=0.10$

The probability distribution function of X is:

$f_{X}(x)=\left \{ {{\lambda e^{-\lambda x};\ x>0}} \atop {0;\ otherwise}} \right.$

Compute the probability that the time between the next two calls is between 3 minutes and 7 minutes as follows:

$P(3$

Thus, the probability that the time between the next two calls is between 3 minutes and 7 minutes is 0.2442.