1. CS723 - Probability and Stochastic Processes

2. Lecture No. 10

3. In Previous Lectures
• Treatment of random experiments with equally likely outcomes • Random Variables • Outcomes get mapped (many to one) to real line and probability is define from • real line to points on [0,1] • For finite sample space, the random variable can only assume finite number of values • For any random variable, the sum of all probabilities is equal to 1

4. Sum of Two Dice

5. CDF of Poisson PMF

6. Chuk-a-luck Modified
Three dice with three different colours red, green, and blue are thrown You call a 3-digit vector ( no digit greater than 6) as your lucky number. You win 3 dollars if your chosen vector is exactly the same as (R,G,B) observed If two components of your chosen vector match, you win two dollars, etc. etc.

7. PMF and CDF

8. Prize Bonds
For poor people, bonds worth Rs. 200 For middle-class people, bonds worth Rs. 750 and Rs For rich people, bonds worth Rs. 7500, Rs and Rs ,000,000 bonds in each series and multiple series circulating One jackpot, 3 runner up, and consolation prizes

9. A single experiment can be used to define two random variables X and Y Both random variables that their values after every run of the experiment The relationship between two random variables could be of interest We can work with the two random variables with a joint distribution We can work with joint PMF and joint distribution without knowing exp.

10. Joint Distribution: Example
You throw 4 dice, 2 red and 2 green, and define the random variables X and Y X = Sum of dots on two red dice Y = abs (Difference of dots on green dice) X takes 11 and Y takes 6 possible values Domain of joint random variables is a 11x6 grid on 2-D plane PMF and CDF are Z 2 → R functions Events can involve X, Y, or both

11. Joint Distribution: Example2 5 4 6 8 10 12 16 20 24 3 18 30 36 32 40 48 1 50 60 pY 7 9 11 pX

12. Joint Distribution: Example
You throw 2 identical dice and define the random variables X and Y X = Sum of dots on two dice Y = abs (Difference of dots two dice) X takes 11 and Y takes 6 possible values Domain of joint random variables is a 11x6 grid on 2-D plane PMF and CDF are Z 2 → R functions Events can involve X, Y, or both

13. Joint Distribution: Example2 5 4 6 3 8 10 1 pY 7 9 11 12 pX

14. Independence of X and Y
Check for independence of X and Y through unconditional and conditional probabilities of events For example Pr(X=5) vs. Pr(X=5 | Y=2) Pr(X=5 | Y =2) = Pr(X=5,Y=2) / Pr(Y=2) = (32/1296)/(8/36) = 4/36 = Pr(X=5) Hence, X and Y are independent

15. Marginal PMF
Get PMF of one random variable from the joint PMF of two RV’s Given the joint PMF of random variables X and Y as fXY(i,j) = Pr(X=i, Y=j) fY(y1) = Pr(Y=y1) = ∑ Pr(X=i, Y=y1)