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FrozenT [24]
3 years ago
11

The time between calls made to Amazon's customer service is exponentially distributed with a mean of 10 minutes (taken between e

ach 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
Mathematics
2 answers:
Dmitriy789 [7]3 years ago
7 0

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.

<em>Let X = time between calls made to Amazon's customer service</em>

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

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

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

daser333 [38]3 years ago
3 0

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 <em>X</em> = time between calls made to Amazon's customer service.

The average time between calls is, <em>β</em> = 10 minutes.

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

The probability distribution function of <em>X</em> 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.

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