Random Variables and their PDFs
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This activity helps you understand what happens when random variables are dependent on each other. It goes slightly beyond the AP curriculum.
Open up the Fathom file couples.ftm. (You've met this data file before!) You can find it in the data file that came with Fathom. Or you can type this URL using the "Import URL" command found under the file menu: http://inspire.stat.ucla.edu/unit_02/couples.txt
The data consist of 30 randomly selected couples. Suppose we now select one of these couples at random. Let X = height of the husband.
a) What's the pdf of X?
b) What's the mean and SD of X?
Let Y be the height of the wife in the randomly selected couple.
c) Find the mean and SD of Y.
d) If X and Y were indpt, what would the SD of X -Y be?
e) In Fathom, create a new variable: diffheight. (To do this, open the Inspector. The last row, under the "Attributes" column, says "<new>". Click on this and type diffheight. Then double click in the formula area and define the variable to be H_Height - W_Height.)
f) What's the SD of diffheight?
g) You can find the correlation between husband and wife by dragging a Summary Table onto the workspace. Then drag and drop H_Height into the left side of the table and W_Height into the right. Make sure the Summary table is selected. Then, under the Summary menu item, choose "Add Formula". When the formula box appears, type "correlation()".
h) Compute Var(H_Height) + Var(W_Height) - 2*correlation*SD(H_Height)*SD(W_Height). Take the square root. Is this closer to the SD you found in (f)?
X and Y are correlated, and so the rule Var(X-Y) = Var(X) + Var(Y) doesn't apply. In fact, Var(X-Y) = Var(X) + Var(Y) -2*r*SD(X)*SD(Y). The quantity r*SD(X)*SD(Y) is called the covariance. It measures the variation that X and Y "share", after accounting for the variation they have by themselves.
Note that Var(X + Y) = Var(X) + Var(Y) + 2* SD(X) * SD(Y). If you add random variables, the covariance term is positive. If you subtract it is negative.