1. Functions of Random Variables

2. Methods for determining the distribution of functions of Random Variables
Distribution function method
Moment generating function method
Transformation method

3. Distribution function method
Let X, Y, Z …. have joint density f(x,y,z, …)
Let W = h( X, Y, Z, …)
First step
Find the distribution function of W
G(w) = P[W ≤ w] = P[h( X, Y, Z, …) ≤ w]
Second step
Find the density function of W
g(w) = G'(w).

4. Use of moment generating functions
Using the moment generating functions of X, Y, Z, …determine the moment generating function of W = h(X, Y, Z, …).
Identify the distribution of W from its moment generating function
This procedure works well for sums, linear combinations etc.

5. Some Useful Rules
Let X be a random variable with moment generating function mX(t). Let Y = bX + a
Then mY(t) = mbX + a(t)
= E(e [bX + a]t) = eatmX (bt)
Let X and Y be two independent random variables with moment generating function mX(t) and mY(t) .
Then mX+Y(t) = mX (t) mY (t)

6. M. G. F.’s - Continuous distributions

7. M. G. F.’s - Discrete distributions

8. The Transformation Method
Theorem
Let X denote a random variable with probability density function f(x) and U = h(X).
Assume that h(x) is either strictly increasing (or decreasing) then the probability density of U is:

9. The Transfomation Method (many variables)
Theorem
Let x1, x2,…, xn denote random variables with joint probability density function
f(x1, x2,…, xn )
Let u1 = h1(x1, x2,…, xn).
u2 = h2(x1, x2,…, xn). 
un = hn(x1, x2,…, xn).
define an invertible transformation from the x’s to the u’s

10. Then the joint probability density function of u1, u2,…, un is given by:
where
Jacobian of the transformation

11. The probability of a Gamblers ruin

12. Suppose a gambler is playing a game for which he wins 1$ with probability p and loses 1$ with probability q.
Note the game is fair if p = q = ½.
Suppose also that he starts with an initial fortune of i$ and plays the game until he reaches a fortune of n$ or he loses all his money (his fortune reaches 0$)
What is the probability that he achieves his goal? What is the probability the he loses his fortune?

13. Let Pi = the probability that he achieves his goal?
Let Qi = 1 - Pi = the probability the he loses his fortune?
Let X = the amount that he was won after finishing the game
If the game is fair
Then E [X] = (n – i )Pi + (– i )Qi
= (n – i )Pi + (– i ) (1 –Pi ) = 0
or (n – i )Pi = i(1 –Pi )
and (n – i + i )Pi = i

14. If the game is not fair
Thus or

15. Note
Also

16. hence or
where

17. Note
thus
and

18. table

19. A waiting time paradox

20. Suppose that each person in a restaurant is being served in an “equal” time.
That is, in a group of n people the probability that one person took the longest time is the same for each person, namely
Suppose that a person starts asking people as they leave – “How long did it take you to be served”.
He continues until it he finds someone who took longer than himself
Let X = the number of people that he has to ask.
Then E[X] = ∞.

21. Proof
= The probability that in the group of the first x people together with himself, he took the longest

22. Thus
The harmonic series

23. The harmonic series