Download Lab 7

MATH 214 Lab 5: Continuous Random Variables
In this lab we will examine continuous distributions and random samples from these distributions. The two
we will be looking at are the normal distribution and the exponential distribution.
Label the columns in your Minitab worksheet as follows:
Old Name
C1
C2
C3
C4
C5
C6
New Name
N Sample 50
N Sample 100
N Sample 10000
E Sample 50
E Sample 100
E Sample 10000
I. The Normal Distribution
A. This distribution has the probability density function
f x  
1
 x 
 0 .5 

  
2
e
,
 2
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.
Graph the 4 normal curves together that all have mean 0 and their standard deviations are 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 window
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.
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 yvalue the function with standard deviation 5 takes on?
Can you imagine a distribution of a random variable with negative standard deviation? With deviation
0? What the zero deviation would mean for a random variable?
Now graph the 4 normal curves together that all have standard deviation 1 and their means are -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 window
Choose 1 in Standard Deviation
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.
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 zero mean and standard deviation 1
Check the radio button X-value and double click on Middle
For x values choose -5 and 5
Click OK, OK.
Repeat this procedure 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 the random sample of size 50 in the column N Sample.
Click Calc, Random Data, Normal
Type 50 in Generate Rows of Data window
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
repeating the procedure above. Make sure you change the number of data rows to generate from 50 to 100 and
10000.
3. Construct a histogram for the three samples you generated. From the histogram menu, select
“with fit”. On the next screen click “data view” and open “distribution” tab. Select normal distribution with
mean 0 and standard deviation 1.
Compare the histograms of the 3 samples to the graph of the theoretical distribution.
(The blue curve on the graph corresponds to the theoretical standard normal distribution.)
How well do the histograms resemble the theoretical distribution?
What happens to the empirical histograms as sample size increases?
II. The Exponential distribution.
A. This distribution has the density function f x  
1
e
x

, 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 mean θ = 1 and 5.
Click Graph, Probability Distribution Plots
Double click on Vary Parameters
Choose Exponential Distribution, type 1 5 in Scales window
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 the graph changes as you change the scale (mean)?
There is one other parameter in 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 Scales window
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 same X and same Y scale
Click OK, OK.
Compare the graphs. What the threshold parameter is responsible for?
B. Select a random sample from the exponential distribution.
1. Store the 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. Make sure you change the number of data rows to generate from 50 to 100 and 10000.
3. Construct a histogram for the three samples you generated. From the histogram menu, select “with
fit”. On the next screen click “data view” and open “distribution” tab. Select exponential distribution with
mean 1.
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 sample size
increases?
III. Applications.
For each random variable family (normal and exponential) in today’s lab, find (in the book or make one up)
a realistic example of such a variable. Also use Minitab to calculate a probability for your example of the
form: P(a < X < b), and interpret the probability. Specify the parameters of your distributions and values of
a and b.