Label the Columns in Your Minitab Worksheet As Follows

MATH 214 Lab 7: Continuous Random Variables

In this lab we will examine probability distributions of continuous random variables and random samples from these distributions. The two we will look at are the normal distribution and the exponential distribution.

Label the columns in your Minitab worksheet as follows:

1. The Normal Distribution

1. This distribution has the probability density function

,

where μ is the mean and σ is the standard deviation. First we will use MINITAB to get a feel for how changing μ and σ affects the shape of the theoretical probability density function.

First, graph four normal curves on the same graph, each with mean 0 and standard deviations 1, 3, 5, and 10 respectively.

Click Graph, Probability Distribution Plots

Double click on Vary Parameters

Choose Normal Distribution with mean 0 and type 1 3 5 10 in Standard Deviation field

Click Multiple Graphs, check the radio button “In Separate Panels of the same graph”

Click OK, OK.

Describe what happens to the normal distribution as you increase the standard deviation.

What is the maximum y-value the function with standard deviation 1 takes on? What is the maximum y-value the function with standard deviation 5 takes on?

What would the distribution of a random variable whose standard deviation is 0 look like? What would it mean if the standard deviation was 0 for a random variable?

Now graph five normal curves on the same graph, each with standard deviation 1 and means -1, -3, 0, 5 and 10 respectively.

Click Graph, Probability Distribution Plots

Double click on Vary Parameters

Choose Normal Distribution, type -1 - 3 0 5 10 in Mean field

Choose 1 in Standard Deviation

Click Multiple Graphs, check the radio button “In Separate Panels of the same graph”

Click OK, OK.

Describe what happens to the normal distribution as you change the mean.

At this point we will verify that the normal curve is a probability density function. A standard normal random variable can take values from negative infinity to positive infinity, but recall that values outside of 3 standard deviations away from the mean are very unlikely.

Click Graph, Probability Distribution Plots

Double click on View Probability

Choose Normal Distribution with mean 0 and standard deviation 1

Click the “Shaded Area” tab

Check the radio button X-value and click on Middle

For x values choose -5 and 5

Click OK, OK.

Use the procedure described above to find the probability that a normal random variable with mean 68 and standard deviation 2.5 takes values between 64 and 70.

B. Select a random sample from the standard normal distribution.

1. Store a random sample of size 50 in the column N Sample 50.

Click Calc, Random Data, Normal

Type 50 in the “Number of rows of data to generate” field

Click in Store in Columns Box

Double Click N Sample 50

Click OK

2. Generate a random sample from the standard normal distribution of size 100 and 10000 using the procedure above.

3. Construct a histogram for the three samples you generated. Display your histograms in separate panels of the same graph using the same x scale.

Compare the histograms of the 3 samples to the graph of the theoretical distribution. How well do the histograms resemble the theoretical distribution? What happens to the empirical histograms as the sample size increases?

1. The Exponential distribution.

1. This distribution has the density function, for x > 0, where θ is the mean and standard deviation.

Considering the formula, NOT THE GRAPH, of this function, what do you expect happens to the value of f(x) as x → ∞?

Generate graphs of the probability density function for the exponential distribution with means θ = 1 and 5.

Click Graph, Probability Distribution Plots

Double click on Vary Parameters

Choose Exponential Distribution, type 1 5 in the Scales field

Click Multiple Graphs, check the radio button “In Separate Panels of the same graph”

Make sure you choose same X and same Y scale

Click OK, OK.

Compare the graphs. How does the graph change as you change the scale (mean)?

There is one other parameter of the exponential distribution which is easy to study with the help of MINITAB:

Click Graph, Probability Distribution Plots

Double click on Vary Parameters

Choose Exponential Distribution, type 1 in the Scales field

Choose two different values for the threshold

Click Multiple Graphs, check the radio button “In Separate Panels of the same graph”

Make sure you choose the same X and same Y scale

Click OK, OK.

Compare the graphs. What effect does the threshold parameter have on the graph?

B. Select a random sample from the exponential distribution.

1. Store a random sample of size 50 in the column E Sample 50.

Click Calc, Random Data, Exponential

Type 50 in Generate Rows of Data window

Click in Store in Columns Box

Double Click E Sample 50

Click OK

2. Generate a random sample from the exponential distribution of size 100 and 10000 repeating the procedure above.

3. Construct histograms for each of the three samples you generated. Display all three histograms on one graph.

Compare the histograms of the 3 samples to the graph of the theoretical distribution. How well do the histograms resemble the theoretical distribution? What happens to the empirical histograms as the sample size increases?

1. Classifying Data

The data in ContinuousSamples.MTW are samples selected from the following populations:

a. Normal (mean 0, standard deviation 1),

b. Normal (mean 0, standard deviation 0.25),

c. Normal (mean 2, standard deviation 1),

d. Exponential (scale = 1, threshold = 0),

e. Exponential (scale = 3, threshold = 0),

f. Exponential (scale = 3, threshold = 3).

Identify which sample came from which population, and explain your reasoning.

1. Applications.

Find a real life example of a random variable that is modeled by a normal probability distribution. Also find a real life example of a random variable that is modeled by an exponential probability distribution. Clearly describe your example and use Minitab to calculate a probability of the form P(a < X < b) for your example. In addition, interpret the probability; that is, explain to someone who knows no statistics the meaning of the probability you computed. Be sure to state the parameters of your distributions and the values of a and b.