Question

Now let's try a normal distribution. The first plot will be easy, but the second will be more difficult to do, so follow the instructions carefully. First we'll plot a standard normal, then we'll see what happens if you change the parameters. x <- seq (-3.291, 3.291, length.out=100) gives us 100 equally spaced values of x between -3.291 and 3.291. “seq” let's us generate a sequence of numbers, and “length.out” tells us how many numbers we want in the sequence. plot(x, dnorm(x, mean = 0, sd = 1), col = "red", type = "l") “plot” is a command to generate graphs (it's quite versatile, and we can't explain everything here). “dnorm” is essentially the equation for the normal distribution. This command will plot x and the value of x at each of the 100 points using the normal equation. The “mean”, “sd” and “col” functions should be self-explanatory. type = “l” (that's lower case “L”, not “one”) generates lines instead of points (try it without type = “l” to see the difference). At this point you should have a standard normal curve. Now we'll see what happens when we change the parameters. First, let's change the mean from 0 to 1: lines(x, dnorm(x, mean = 1, sd = 1), col = "blue") “lines” will plot lines, but without erasing the previous plot. In other words, it'll plot the lines on the same graph as before. We changed the color to blue so you can see which plot is which. Now answer the following question:

(a) What happened to the normal distribution? How is it different from the first distribution you plotted? You can hold off on copying and pasting your graph until you've done all of the steps below (you only need to hand in one graph for problem 6). Now let's adjust the standard deviation: lines(x, dnorm(x, mean = 0, sd = 2), col = "green") you should know how this works at this point. Answer the following question:

(b) Again, what happened to the normal distribution? How is it different from the previous normal distributions you plotted? Finally, let's adjust both the mean and standard deviation. lines(x, dnorm(x, mean = 1.5, sd = 1.7), col = "purple")

(c) How is this different from the above normal distributions? At this point your graph should have four curves plotted on top of each other in your final graph. Copy this graph and paste it into a word processor or whatever you want to use so that you can print a copy.

(d) Can you understand now how important the parameters of a distribution are? What do the parameters of a distribution tell us about the distribution?

Answer 1

a. This entirely depends on location and scale. When the location is 0 and scale is 1 that is standard normal distribution with colour red. So the mean ,mode ,median of the distribution are at 0 and it is symmetric about the location 0.

From blue curve we observe both the location has changed from 0 to 1, that is now the mean, median and mode are 1 and the variance is still 1. So the curve shifts to right because now it is symmetric about 1. But the shape of the curve remains same that is it neither becomes wider nor narrower.

b.when scale changes from 1 to 2, the location remains same, that is mean,median and mode are still at 0 but the deviation increases, that is now there are less number of observations from (-3,3) and they have become more distributed over the X axis which is evident from green curve which is wide.

c. From the purple curve we see that mean increases from 0 to 1.5 and so the location shifts right again( more than 1). Also the deviation increases to 1.7 so it becomes wider than red and blue curve but narrower than green curve which has standard deviation 2.

d. So the location parameter shifts the graph and the scale parameter defines the stretch of the graph. This way the parameters of the distribution characterize the shape of the distribution.