1. Chapter 5.1 & 5.2: Random Variables and Probability Mass Functions
Chris Morgan, MATH G160
February 1, 2012
Lecture 10

3. Random Variables
A random variable (RV) is a real valued function whose domain is a sample space.
- We will usually denote random variables by X, Y, Z and their respective values by x, y, z

4. Random Variables Discrete Random Variables
•Random variables that take on a finite (or countable) number of values.
–Sum of two dice (2,3,4,…,12)
–Number of children (0,1,2,…)
–Number in attendance at the movies
–Number of hired employees
- Number of students coming to class
Continuous Random Variables
•Random variables that take on values in a continuum or infinitely many values.
–Height
–Weight
–Time
Time you can hold your breath
Lifetime of your cell phone batter

5. Random Variables X P(x) 0 1/16 1 4/16 2 6/16 3 4/16 4 1/16
Toss a fair coin 4 times. Suppose we are interested in the random variable X = number of heads.
Outcome Combination
T T T T C0
T T T H C1
T H H T 4C2
T H H H 4C3
H H H H C4
X P(x)
0 1/16
1 4/16
2 6/16
3 4/16
4 1/16

6. Probability Mass Function (PMF)
The values on the right of the table above are called the Probability Mass Function of the random variable X :
The probability of x = the probability(X = one specific x)
A probability mass function (p.m.f) is a function which describes the probabilities a discrete type random variable will take on for any given value. They can be used to calculate the probabilities corresponding to an event relating to a random variable.

7. Probability Mass Function (PMF)
We can take the PMF, graph it, and display it in the form of a histogram.

8. Probability Mass Function (PMF)x
f(x) 2
1/36 3
2/36 4
3/36 5
4/36 6
5/36 7
6/36 8 9 10 11 12
P(X < 4) =
P(3 < x < 8) =
P(3 ≤ x ≤ 8) =

9. Probability Mass Function (PMF)x
f(x) 2
1/36 3
2/36 4
3/36 5
4/36 6
5/36 7
6/36 8 9 10 11 12
P(x < 8 | x < 10) =
P(x > 3) =
P(4 < x < 7 | x < 9) =

10. Basic Properties of a PMF
(PMFs are nonnegative)
2. There are only finitely (or countably infinitely many) x’s for which: 3.

11. Fundamental Probability Formula
How do you compute probabilities for a random variable X?
Welll….. We have the PMF that tell us the probability that the random variable X takes on the specific value x. Sometimes, you may be interested in a whole range of possible values of X.
For instance, the Pr(1 ≤ x≤ 3)

12. Example
I toss a fair coin 4 times. What the probability of getting at most two most?
P(x ≤ 2) = ?
P(x ≥ 1) = 1 – P(x=0) = 1 - 1/16 = 15/16
X P(x)
0 1/16
1 4/16
2 6/16
3 4/16
4 1/16

13. Example
I can also write this PMF as a function rather than a chart:
X P(x)
0 1/16
1 4/16
2 6/16
3 4/16
4 1/16
For x = 0
For x = 1
For x = 2
For x = 3
For x =
otherwise

14. Fundamental Probability Formula
Suppose X is a discrete RV and that A is a set of real numbers. Then:
In words: the sum of the probability mass function over all possible value you are interested in

15. Practice Problem #1
A partially eaten bag of M&M’s contains 2 red, 5 blue, and 3 green M&M’s. You and your buddy decide to place a bet.
You will choose two M&M’s at random without replacement. For every red M&M you win $5, for every green M&M you win $1 and you do not win anything for a blue M&M.
Let X be the amount of money you will win.

16. Practice Problem #1 [2 red, 5 blue, and 3 green M&M’s….pick two]
Let X be the amount of money you will win. Begin by writing down the PMF:
Colors X
p(x)
R and R 10
1/45
R and G 6
2*3 = 6/45
R and B 5
2*5 = 10/46
B and G 1
5*3 = 15/45
B and B
5C2 = 10/45
G and G 2
3C2 = 3/45

17. Practice Problem #1
What is the probability you will win at least five dollars?
P(X ≥ 5) = 1/45 + 6/ /45 = 17/45
Colors X
p(x)
R and R 10
1/45
R and G 6
2*3 = 6/45
R and B 5
2*5 = 10/46
G and G 2
3C2 = 3/45
B and G 1
5*3 = 15/45
B and B
5C2 = 10/45

18. Practice Problem #1
If you win something, what is the probability it will be worth at least five dollars??
P(X ≥ 5 | X ≠ 0) =

19. Practice Problem #1b E(X)= E(X)2= E(X2)= X p(x) 10 1/45 6 6/45 5 10/46
15/45
10/45 2
3/45

20. Practice Problem #2a
In a simple game, two fair coins are tossed and the payoff is to be determined from the outcome. The payoff strategy is as follows:
Win $5 for each head, lose $10 for two tails.
Let X denote your winnings if you play this game once.

21. Practice Problem #2a Win $5 for each head, lose $10 for two tails.
Write out the PMF for x:
X p(x)
2 heads /4
1 tail, 1 head 5 1/2
2 tails ¼
E(X) = 10(1/4) + 5(1/2) – 10(1/4) = 2.5

22. Practice Problem #2b
In a simple game, two fair coins are tossed and the payoff is to be determined from the outcome. The payoff strategy is as follows:
Lose $5 for two tails
Win $5 for two different
Lose $10 for two heads
Let X denote your winnings if you play this game once.

23. Practice Problem #2b Calculate the PMF for this payoff strategy:
Calculate the expected payoff 
E(X) =
Which payoff would you rather play with, a or b, and why??