Tutorial 5

1. Tutorial 5 Generating Functions & Sum of Independent Random Variables

2. Generating Functions • Generating functions are tools for studying distributions of R.V.’s in a different domain. (c.f. Fourier transform of a signal from time to frequency domain) • Moment Generating Function gX (t)=E[etX] • Ordinary Generating Function hX (z)= E[ZX] • o.g.f. is also called z-transform which is applied to discrete R.V’s only.

3. z-transform • We illustrate the use of g.f.’s by z-transform: • Let a non-negative discrete r.v. X with p.m.f. {pk, k = 0,1,…}, z is a complex no. • The z-transform of {pk} is hX(z) = p0 + p1z + p2z2+ …… =  pkzk • It can be easily seen that  pkzk = E[zX]

4. z-transform • We can obtain many useful properties of r.v. X from hX(z). • First, we can observe that • hX(0) = p0 + p10+ p202+ …… = p0 • hX(1) = p0 + p11+ p212+ …… = 1 • By differentiate hX(z), we can get the mean and variance of X.

5. Mean by z-transform • Put z = 1, we get •  hX’(1) is the mean of of X. • Similarly,

6. Variance by z-transform • E[X2] is called the 2nd moment of X. • In general, E[Xk] is called the k-th moment of X. We can get E[Xk] from successive derivatives of hX (z). • Since Var(X) = E[X2] - E[X]2, we get

7. Example - Bernoulli Distr. • Find the mean and variance of a Bernoulli distr. by z-transform. P(X=1) = p, P(X=0) = 1-p

8. Example - Bernoulli Distr. • E[X] = hX’(1) = p

9. Finding pj from g(t) and h(z) • If we know g(t), then we know h(z), then we can find the pj :

10. p.d.f. of sum of R.V.’s • Let X , Y be 2 independent continuous R.V.’s • The cumulative distribution function (c.d.f) of X+Y:

11. p.d.f. of sum of R.V.’s • By differentiating the above equation, we obtain the p.d.f. of X+Y: • fX+Y(a) is the convolution of fX and fY .

12. m.g.f. of sum of R.V.’s • On the other hand, the moment generating function of p.d.f. fX is • The m.g.f. of fX+Y is:

13. m.g.f. of sum of R.V.’s • We have obtained an important property: • If S = X+Y, where X & Y are independent. • In general, if p.d.f. m.g.f.

14. Two-Armed Bandit Problem • You are in a casino and confronted by two slot machines. Each machine pays off either one dollar or nothing. The probability that the first machine pays off a dollar is x and that the second machine pays off a dollar is y. We assume that x and y are random numbers chosen independently from the interval [0,1] and unknown to you. You are permitted to make a series of ten plays, each time choosing one machine or the other.

15. Two-Armed Bandit Problem • How should you choose to maximize the number of times that you win? • Strategies described in Grinstead and Snell(P.170): • Play-the-best (calculate the prob. that each machine will pay off at each stage and choose the machine with the higher prob. ) • Play-the-winner (choose the same machine when we win and switch machines when we lose)

16. Coursework 01 • Modified two-armed bandit problem: both unknown prob. vary in a linear manner over the twenty plays, Pr(payoff at kth play for machine i) = ai + kbi where ai and bi are constants. • Make a series of 20 plays • Design a simple strategy to maximize the number of times that you win