PMF and Examples

1. PMF and Examples

2. PMF • We have introduced the concept of PMF. • 1. It is short for “Probability Mass Function”. • 2. It is used for discrete random variables. • 3. It specifies the probability of each sample point in the sample space of a random variable. • 4. For each sample point, xi, in the sample space, p(xi) is non-negative • 5. The sum of all p(xi) should be 1.

3. PMF • Form of PMF

4. Example 1 • Roll a 6-sided fair die and let the random variable X be the outcomes of rolling. • Sample space: {1, 2, 3, 4, 5, 6} • PMF:

5. Example 2 • Roll a 6-sided fair die once and let random variable Y be the outcome that whether we get a number of greater than 2.

6. Summary • In order to find the PMF, we need to know • 1. What experiment we are talking about • 2. How the random variable is defined • 3. Find the sample space and corresponding probability

7. Example 3 • 4 players are playing a deck of 52 cards and let X be the number of aces one player could have, find the PMF of X.

8. Example 4 • Homer is playing in a game which has two parts. He must shoot at a target first. If he hits the target, he will then be asked a 5 choice multiple choice question. If Homer hits the target, he will be rewarded $20 and if he gets the question, he will be rewarded $40. Assuming that Homer can hit the target with 60% chance and has no clue about the question, let X be the possible pay-out Homer can get from the game, find the PMF of X.

9. Example 4 • 1. Possible outcomes for Homer from the game {make $60, make $20, get nothing} • 2. Sample space {60, 20, 0} • 3. P(Homer got 0)=P(Homer missed the target) • 4. P(Homer got $20)=P(Homer hit the target but got the question wrong) • 5. P(Homer got $60)=P(Homer hit the target and got the question correctly)

10. Example 4 • Let A={Homer hit the target} and B={Homer got the question correctly} • Then • P(0)=P(Ac)=1-P(A) • P(20)=P(ABc)=P(Bc|A)P(A) • P(60)=P(AB)=P(B|A)P(A)

11. Example 4 • Finally, the PMF is

12. Example 5 • Bart is playing with a fair coin. He decides he will stop until he sees the first head or three tails. Let X be the number of tosses Bart will make and find the PMF of X.

13. Example 5 Possible values of X: {1, 2, 3} P{X=1} P{X=2} P{X=3}

14. Example 5 • PMF of X

15. Find PMF on transforms of X • Given the PMF of X, what is the PMF of 2x+1?

16. Find PMF on transforms of X • Since there is a one-to-one correspondence between X and Y, if we know X, we know Y automatically. The probability that X=xi is exactly the same as the probability that Y=2xi+1

17. Find PMF on transforms of X • What if we are looking for the PMF for Z=X^2? • Similarly, if we know X, we know exactly what Z is here, so we can re-construct the PMF chart in the form of

18. Find PMF on transforms of X • How about the PMF for Z=X^2 if the PMF of X is like the one on the right?

19. Find PMF on transforms of X • In this case, there are two X values that will give the same Z value, (-2)^2=2^2=4. • Therefore, we will need to merge some of the probabilities to create the PMF chart • The PMF should look a little different.

20. Example 6 • The PMF of X is given and we want to find the PMF of Z=X^2 • First, we want to verify it is a valid PMF, add up all X’s to check whether it is 1.

21. Example 6 • Z=X^2, therefore, we should have a different set of values for Z and we want to keep track of the probabilities too. • For example, Z=1 if X=-1 with p=1/8 and X=1 with p=3/8; Z=2.25 if X=-1.5 with p=1/16, etc.

22. Find probabilities given PMF • Given a PMF, we can find the following probabilities: • P(X=xi), P(x1<X<x2), P(X>xi) or P(X<xi) • In those cases, we just find all X’s whose values fall within the range and add up the corresponding probabilities.

23. Find Probabilities given PMF • In example 6: let’s find the following probabilities: • 1. P(X>0) • 2. P(-1.8<X<2.5) • 3. P(X<3)