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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.