1. Winter 2006EE384x1 Review of Probability Theory Review Session 1 EE384X

2. Winter 2006EE384x2  A random experiment with set of outcomes  Random variable is a function from set of outcomes to real numbers Random Variable

3. Winter 2006EE384x3 Example  Indicator random variable:  A : A subset of is called an event

4. Winter 2006EE384x4 CDF and PDF  Discrete random variable:  The possible values are discrete (countable)  Continuous random variable:  The rv can take a range of values in R  Cumulative Distribution Function (CDF):  PDF and PMF:

5. Winter 2006EE384x5 Expectation and higher moments  Expectation (mean):  if X>0 :  Variance:

6. Winter 2006EE384x6 Two or more random variables  Joint CDF:  Covariance:

7. Winter 2006EE384x7 Independence  For two events A and B :  Two random variables  IID : Independent and Identically Distributed

8. Winter 2006EE384x8 Useful Distributions

9. Winter 2006EE384x9 Bernoulli Distribution  The same as indicator rv:  IID Bernoulli rvs (e.g. sequence of coin flips)

10. Winter 2006EE384x10 Binomial Distribution  Repeated Trials:  Repeat the same random experiment n times. (Experiments are independent of each other)  Number of times an event A happens among n trials has Binomial distribution  (e.g., number of heads in n coin tosses, number of arrivals in n time slots,…)  Binomial is sum of n IID Bernouli rvs

11. Winter 2006EE384x11 Mean of Binomial  Note that:

12. Winter 2006EE384x12 Binomial - Example n=4 n=40 n=10 n=20 p=0.2

13. Winter 2006EE384x13 Binomial – Example (ball-bin)  There are B bins, n balls are randomly dropped into bins.  : Probability that a ball goes to bin i  : Number of balls in bin i after n drops

14. Winter 2006EE384x14 Multinomial Distribution  Generalization of Binomial  Repeated Trails (we are interested in more than just one event A )  A partition of  into A 1,A 2,…,A l  X i shows the number of times A i occurs among n trails.

15. Winter 2006EE384x15 Poisson Distribution  Used to model number of arrivals

16. Winter 2006EE384x16 Poisson Graphs =10 =.5 =1 =4

17. Winter 2006EE384x17 Poisson as limit of Binomial  Poisson is the limit of Binomial(n, p) as  Let

18. Winter 2006EE384x18 Poisson and Binomial Poisson(4) n=5,p=4/5 n=20, p=.2 n=10,p=.4 n=50,p=.08

19. Winter 2006EE384x19 Geometric Distribution  Repeated Trials: Number of trials till some event occurs

20. Winter 2006EE384x20 Exponential Distribution  Continuous random variable  Models lifetime, inter-arrivals,…

21. Winter 2006EE384x21 Minimum of Independent Exponential rvs  : Independent Exponentials

22. Winter 2006EE384x22 Memoryless property  True for Geometric and Exponential Dist.:  The coin does not remember that it came up tails l times  Root cause of Markov Property.

23. Winter 2006EE384x23 Proof for Geometric

24. Winter 2006EE384x24  Moment Generating Function (MGF)  For continuous rvs (similar to Laplace xform)  For Discrete rvs (similar to Z-transform): Characteristic Function

25. Winter 2006EE384x25 Characteristic Function  Can be used to compute mean or higher moments:  If X and Y are independent and T=X+Y

26. Winter 2006EE384x26 Useful CFs  Bernoulli( p ) :  Binomial( n, p ) :  Multinomial :  Poisson :