1. Discrete Random Variables 2

2. Random Variable
Numerical attribute of an experimental outcome.

3. Probability Mass Function (PMF)

4. Functions of Random Variables
Y = 4*H3 + 75
Y = H – E(H)
Y = if H = 0
0 if H >= 1

5. Weighted average of all possible outcomes.
Expectation
Weighted average of all possible outcomes.
E[x] = ∑ [ x px(x) ]
Variance
Measures the spread of the PMF around the expected value. or
Y = (X – E(X))2
σx2 = E(Y) As
This is why PMF probability mass function – if you think of probability as a physical weight – and the x axis as the position, then computing E[x] is like computing where the center of mass will be on the x axis

6. Functions of Random Variables (cont)
Y = if h <= 1
0 if h >= 2
E(H) = 1.5
E(Y) = ?
In general, for any var. X
and func. g(X):
if Y = g(X): E(Y) = ∑ [ g(X) px(X) ]

7. Bernoulli Random Variable
Experiment: Toss coin once
1-p
0.5 p H
0.5 T X
0 (T)
1 (H)
Examples of experiments with 2 possible outcomes:
- is a person healthy or sick?
- do you like a song on pandora.com?
- will event A occur or not?
P(X=1) = p
P(X=0) = 1-p

8. Bernoulli Random Variable (cont)p
Experiment: Toss coin once
PMF:
1-p X
E(X) = p
0 (T)
1 (H)
variance(X) =
p(1-p) 1
CDF:
1-p X
0 (T)
1 (H)

9. Binomial Random Variable
Experiment:
number of tosses: 4
probability of heads: ¾
X = number of heads H
HHHH H T
HHHT
P(X=2) ?
6 x (¾)2 x (¼)2 = 27/128 = 0.21 H
P(X=3) ?
4 x (¾)3 x (¼) = 108/256 = 0.42 H
HHTH T T
HHTT H H
HTHH H T
HTHT T H
HTTH T T
HTTT
Generalized Experiment:
number of tosses: n
probability of heads: p H
THHH H T
THHT H H
THTH T T
THTT T
P(X = k) = ? H
TTHH H T
TTHT
(n C k) pk (1-p)n-k T H
TTTH T T
TTTT

10. Binomial Random Variable (cont)
PMF:
E(X) = np
Variance(X) = np(1-p) k
CDF: k

11. Geometric Random Variable
Experiment:
number of tosses: 3
probability of heads: ¾
X = number of tosses until you get heads
P(X=3) = ?
P(H1) = 48/64
P(T1 H2) = 12/64
P(T1 T2 H2) = 3/64 H1 H2 H3
P(H1) = 3/4
P(H2|T1) = 3/4
P(H2|T1 T2) = 3/4 T1 T2 T3
P(T1) = 1/4
P(T2 | T1) = 1/4
P(T3 | T1 T2) = 1/4
Generalized Experiment:
number of tosses: n
probability of heads: p
P(X = k) = ?

12. Geometric Random Variable (cont)
PMF:
E(X) =
Variance(X) =
CDF:

15. Independence of Random Variables

16. Some thoughts What does it mean for 2 experiments to be independent?
How do you derive the properties of binomial random variables from
Bernoulli random variables?

17. Other topics:
- PMFs of more than one random variable
- conditional PMF