STA230/MTH230: Probability

Homework #7: Density Functions & CDF's

Text Problems are from Jim Pitman, Probability.

• Exercises
  

  § 4.1:		2	3	6	8
  § 4.2:		1	5	8	13
  § 4.3:		3

• Another Problem
  Let U have the uniform distribution on the interval (0,1) and define new random variables X=√U and Y=1/X=1/√U. The object is to find the mean μ, variance σ², CDF, and pdf for X and Y.

  Calculate E[X], E[X²], E[Y], and E[Y²]. From these, find the mean and variance of X and Y. (Suggestion: calculate the expected power E[U^p] for all real numbers p, then pick off the four special cases needed above).

  Find the CDFs F_x(x)=P[X≤x] and F_y(y)=P[Y≤y] by expressing the events "[X≤x]" and "[Y≤y]" in terms of U. First think through what are the possible values of X and Y, as U ranges from zero to one.

  Finally, evaluate and graph the pdf functions f_x(x) and f_y(y) by differentiating. Be sure to give the correct values for all x and y in (-∞,+∞).

  The random variable X has a special case of the Beta distribution, often used to model uncertain probabilities or proportions, and Y has a special case of the Pareto distribution, often used to model quantities like incomes with extremely wide ranges or "heavy tails". I hope you have a surprise or two and have infinitely much fun with this!

Good Luck!