PMF and Examples

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.

PMF Form of PMF

X x1 x2 x3 … xn

P(X) P(x1) P(x2) P(x3) … P(xn)

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:

X P(X)

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

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.

X P(X)

1 (greater than 2)

4/6 or 2/3

0 (less than or equal 2)

2/6 or 1/3

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

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.

X P(X)

0

1

2

3

4

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.

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)

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)

Example 4 Finally, the PMF is X P(X)

0 0.4

20 0.48

60 0.12

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.

Example 5

Possible values of X: {1, 2, 3}P{X=1}P{X=2}P{X=3}

Example 5 PMF of X X P(X)

1 1/2

2 1/4

3 1/4

Find PMF on transforms of X Given the PMF of

X, what is the PMF of 2x+1?

X P(X)

1 1/2

2 1/4

3 1/4

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

X Y P(Y)

1 3 1/2

2 5 1/4

3 7 1/4

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

X Z P(Z)

1 1 1/2

2 4 1/4

3 9 1/4

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?

X P(X)

-2 1/2

1 1/4

2 1/4

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.

Z P(Z)

1 1/4

4 3/4

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.

X P(X)

-2 1/8

-1.5 1/16

-1 1/8

0 1/4

1 3/8

2 1/32

3 1/32

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.

Z P(Z)

0 1/4

1 1/2

2.25 1/16

4 5/32

9 1/32

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.

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)