# EXERCISES

• 1. Derive Bayes’ theorem.
• 2. Show that Bayes’ theorem is true even if the two events are independent.
• 3. Show that the geometric distribution is correctly normalized (i.e., the sum of all possible observations is 1). (Hint: Use the classical

an = 1/(1 - a).)

n =0

• 4. It’s not quite true that -^ to ^ is the only range for which the Gaussian integral can be evaluated. There is one other range. What is it, and what’s the value of the integral? (Hint: No calculations required—just think about the shape of the Gaussian distribution.)
• 5. Explain why n + m + 1 divided by 2 is the expected average rank in the

ZN

j = (N (N+1))/2.)

• 6. Use the central limit theorem to argue that the null distribution of the WMW test statistic should be Gaussian.
• 7. Consider the following two lists of numbers. The first list is 1, 2, then ten 3s, then 4 and 5. The second list is the same, except there are one hundred 3s (instead of 10). Draw the cumulative distributions for these two lists, and explain why this might lead to a problem for the KS-test if ties were not handled correctly.
• 8. In the P-values for Gene Set Enrichment Analysis, why does the first sum go up only until min(n, l) and the second one go up to k - 1?
• 9. Look up (or plot) the shapes of the Poisson and Geometric distribution. Speculate as to why the Poisson is used more often.