The length of time in minutes, X, between 911 calls in Metropolis has an exponential distribution with a mean of 20.00 minutes. Thus, the density of X is:



*f*(*x*,𝜆) = 𝜆 *e*−𝜆*x*

 for 0 ≤ x ≤ ∞, 𝜆 is what R calls rate.

Hint: This is a problem involving the exponential distribution. Knowing the parameter 𝜆 for the distribution allows you to easily answer parts a ,b ,c and use the built-in R functions for the exponential distribution (dexp(), pexp(), qexp()) for other parts . Or (not recommended) you should be able to use the R integrate command with f(x) defined as above or with dexp() for all parts.

a) What is the value of 𝜆? 

b) What is the variance of X? 

c) What is the standard deviation of X? 

d) What is the probability that X is greater than its expected value? 

e) What is the probability that X is > 25? 

f) What is the probability that X is > 50? 

g) What is the probability that X > 50 given that X > 25? 

h) What is the median of X? 

l) Enter any comments in the text box below.


To solve this problem involving the exponential distribution, let's go through each part step by step.

### Given Information
The mean of the exponential distribution \( \mu \) is 20.00 minutes. The rate parameter \( \lambda \) is related to the mean by the formula:

\[
\lambda = \frac{1}{\mu}
\]

### a) What is the value of \( \lambda \)?

Using the mean:

\[
\lambda = \frac{1}{20} = 0.05
\]

### b) What is the variance of \( X \)?

The variance \( \sigma^2 \) of an exponential distribution is given by:

\[
\sigma^2 = \frac{1}{\lambda^2}
\]

Substituting the value of \( \lambda \):

\[
\sigma^2 = \frac{1}{(0.05)^2} = \frac{1}{0.0025} = 400
\]

### c) What is the standard deviation of \( X \)?

The standard deviation \( \sigma \) is the square root of the variance:

\[
\sigma = \sqrt{400} = 20
\]

### d) What is the probability that \( X \) is greater than its expected value?

The probability \( P(X > \mu) \) for an exponential distribution is calculated as follows:

\[
P(X > \mu) = P(X > 20) = e^{-\lambda \cdot 20} = e^{-0.05 \cdot 20} = e^{-1} \approx 0.3679
\]

### e) What is the probability that \( X > 25 \)?

Using the exponential distribution's survival function:

\[
P(X > 25) = e^{-\lambda \cdot 25} = e^{-0.05 \cdot 25} = e^{-1.25} \approx 0.2865
\]

### f) What is the probability that \( X > 50 \)?

\[
P(X > 50) = e^{-\lambda \cdot 50} = e^{-0.05 \cdot 50} = e^{-2.5} \approx 0.0821
\]

### g) What is the probability that \( X > 50 \) given that \( X > 25 \)?

Using the memoryless property of the exponential distribution:

\[
P(X > 50 | X > 25) = P(X > 50 - 25) = P(X > 25) = e^{-0.05 \cdot 25} \approx 0.2865
\]

### h) What is the median of \( X \)?

The median \( M \) of an exponential distribution can be calculated as:

\[
M = \frac{\ln(2)}{\lambda}
\]

Calculating:

\[
M \approx \frac{0.6931}{0.05} \approx 13.86
\]

### Summary of Answers
- a) \( \lambda = 0.05 \)
- b) Variance \( \sigma^2 = 400 \)
- c) Standard Deviation \( \sigma = 20 \)
- d) \( P(X > 20) \approx 0.3679 \)
- e) \( P(X > 25) \approx 0.2865 \)
- f) \( P(X > 50) \approx 0.0821 \)
- g) \( P(X > 50 | X > 25) \approx 0.2865 \)
- h) Median \( \approx 13.86 \)

If you have any questions or need further clarification on any part, let me know! Here are some follow-up questions:

1. How is the exponential distribution applied in real-world scenarios?
2. Can you explain the memoryless property in more detail?
3. What would happen to the probabilities if the mean was decreased?
4. How can we graph the probability density function for this distribution?
5. Are there any other distributions that have similar properties to the exponential distribution?

**Tip:** Remember that the exponential distribution is often used to model time until an event occurs, such as failures or arrivals.

The length of time in minutes, X, between 911 calls in Metropolis has an exponential distribution with a mean of 20.00 minutes. Thus, the density of X is:

f(x,𝜆) = 𝜆 e^(-𝜆 x)

for 0 ≤ x ≤ ∞, 𝜆 is what R calls rate.

Hint: This is a problem involving the exponential distribution. Knowing the parameter 𝜆 for the distribution allows you to easily answer parts a, b, c and use the built-in R functions for the exponential distribution (dexp(), pexp(), qexp()) for other parts. Or (not recommended) you should be able to use the R integrate command with f(x) defined as above or with dexp() for all parts.

a) What is the value of 𝜆?

b) What is the variance of X?

c) What is the standard deviation of X?

d) What is the probability that X is greater than its expected value?

e) What is the probability that X is > 25?

f) What is the probability that X is > 50?

g) What is the probability that X > 50 given that X > 25?

h) What is the median of X?

This problem explores the exponential distribution modeling the time between 911 calls with a mean of 20 minutes. Learn to calculate the rate (𝜆), variance, standard deviation, probabilities of call intervals, and the median. Perfect for advanced high school or college statistics students.

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