Question

A variable is said to have an exponential distribution or to be exponentially distributed if its distribution has the shape of an exponential curve, that is a curve of the form \(y=\frac{e^{\frac{-x}{\mu}}}{\mu}\) for \(x>0\), where \(\mu\) is the mean of the variable. The standard deviation of such a variable also equals \(\mu\). At the emergency room at Desert Samaritan Hospital in Mesa, Arizona, the time from the arrival of one patient to the next, called an interarrival time, has an exponential distribution with a mean of \(8.7\) minuts.

a. Sketch the exponential curve for the distribution of the variable "interarrival time". Note that this variable is far from being normally distributed. What shape does its distribution have?

b. Use the technology of your choice to simulate \(1000\) samples of four interarrival times each.

c. Find the sample mean of each of the \(1000\) samples.

d. Determine the mean and standard deviation of the \(1000\) sample means.

e. Theoretically, what are the mean and the standard deviation of all possible sample means for samples of size \(4\)? Compare your answers to those you obtained in part (d).

f. Obtain a histogram of the \(1000\) sample means. Is the histogram bell shaped? Would you necessarily expect it to be?

g. Repeat parts (b)-(f) for a sample size of \(40\).

Short answer

Part a.

Part b.

Part c. The mean of the sample at different time arrival is mean \(=8.6818\)

Part d. The standard deviation of the sample at different time arrival is \(STD=8.3618\)

Part e. Then calculated mean is slightly less than the actual mean \(=8.6818\)

The standard deviation is greater than the actual standard deviation. \(STD=8.3618\)

Step-by-step solution

Part a. Step 1. Given information

The number of samples and mean is given

\(x=1:15\)

\(\mu=8.7\)

The function has been given

\(y=\frac{1}{\mu}e^{\frac{-x}{\mu}}\)

Part a. Step 2. Calculation

Let's take \(x=1:15\)

The mean is given \(\mu=8.7\)

Put all these value into the given equation

\(y=\frac{1}{\mu}e^{\frac{-x}{\mu}}\)

\(y=\frac{1}{8.7}e^{\frac{-1:15}{8.7}}\)

After solving we will get the answer

Sketch a graph.

Program:

Query:

• First, we have defined the number of samples.
• The define the given mean.
• Write the given equation.
• Put all these value into the given equation and get the value of y.

• Sketch a graph between number of sample and y.

Part b. Step 1. Given information

The number of samples and mean is given

\(x=1:1000\)

\(\mu=2,4,8,10\)

The function has been given

\(y=\frac{1}{\mu}e^{\frac{-x}{\mu}}\)

Interarrival time is \(4\).

Part b. Step 2. Calculation

Let's take \(x=1:1000\)

The mean is given \(\mu=2,4,8,10\)

Put all these value into the given equation

\(y=\frac{1}{\mu}e^{\frac{-x}{\mu}}\)

For \(\mu=2\)

\(y=\frac{1}{2}e^{\frac{-1:1000}{2}}\)

After solving we will get the answer

Sketch a graph.

Program:

Query:

• First, we have defined the number of samples.
• The define the given mean.
• Write the given equation.
• Put all these value into the given equation and get the value of y.
• Sketch a graph between number of sample and y.

Part c. Step 1. Calculation

Let's take \(x=1:1000\)

The mean is given \(\mu=2,4,8,10\)

Put all these value into the given equation

\(y=\frac{1}{\mu}e^{\frac{-x}{\mu}}\)

For \(\mu=8.7\)

\(y=\frac{1}{8.7}e^{\frac{-1:1000}{8.7}}\)

After solving we will get the value of y.

Calculate the mean

\(mean=\frac{\sum_{n-i}^{1000}y}{1000}\)

After solving we will get

\(mean=8.6818\)

Program:

Query:

• First, we have defined the number of samples.
• The define the given mean.
• Write the given equation.
• Put all these value into the given equation and get the value of y.
• Calculate the mean of all the samples at different interarrival time.

Part d. Step 1. Calculation

Let's take \(x=1:1000\)

The mean is given \(\mu=2,4,8,10\)

Put all these value into the given equation

\(y=\frac{1}{\mu}e^{\frac{-x}{\mu}}\)

For \(\mu=8.7\)

\(y=\frac{1}{8.7}e^{\frac{-1:1000}{8.7}}\)

After solving we will get the value of y.

Calculate the standard deviation

\(STD=\sqrt{\frac{\sum_{n-1}^{1000}y_{n}-\mu}{1000}}\)

After solving we will get

\(STD=8.3618\)

Program:

Query:

• First, we have defined the number of samples.
• The define the given mean.
• Write the given equation.
• Put all these value into the given equation and get the value of y.
• Calculate the standard deviation of all the samples at different interarrival time.

Part e. Step 1. Calculation

The given mean is

\(\mu=8.7\)

And after calculation we will get

\(mean=\frac{\sum_{n-1}^{1000}y}{1000}=8.6818\)

The actual standard deviation would be

\(STD=\frac{\mu}{\sqrt{1000}}=4.35\)

But after calculating we will get the standard deviation

\(STD=8.3618\)