1. POPULATION (of “units”)
uniform
X = “Random Variable”
symmetric unimodal
“REAL WORLD” SYSTEM
skew (positive)
PROBABILTY MODEL
YES
Model has to be tweaked.
THEORY
EXPERIMENT
Is there a significant difference?
Random Sample
Model Predictions
STATISTICS
How do we test them? NO
Model may be adequate / useful.

2. POPULATION (of “units”)
uniform
“Arbitrarily large” or infinite collection of units (rocks, toasters, people, districts,…)
X = “Random Variable”
symmetric unimodal
“REAL WORLD” SYSTEM
A numerical value assigned to each unit of the population
skew (positive)
PROBABILTY MODEL
Temp
Mass
Foot length
Shoe size
# children
TV channel
Alphabet
Zip Code
Shirt color
Coin toss
Pregnant?
Quantitative
(A = 01,…, Z = 26)
Qualitative
(Blue = 1, White = 2…)
(Heads = 1, Tails = 0)
(Yes = 1, No = 0)

3. POPULATION (of “units”)
uniform
skew (positive)
symmetric unimodal
“Arbitrarily large” or infinite collection of units (rocks, toasters, people, districts,…)
X = “Random Variable”
“REAL WORLD” SYSTEM
A numerical value assigned to each unit of the population
PROBABILTY MODEL
Temp
Mass
Foot length
Shoe size
# children
TV channel
Alphabet
Zip Code
Shirt color
Coin toss
Pregnant?
Continuous
Quantitative
Discrete
Ordinal
(A = 01,…, Z = 26)
Qualitative
(Blue = 1, White = 2…)
Nominal
(Heads = 1, Tails = 0)
Binary
(Yes = 1, No = 0)

4. POPULATION (of “units”)
uniform
skew (positive)
symmetric unimodal
“Arbitrarily large” or infinite collection of units (rocks, toasters, people, districts,…)
X = “Random Variable”
“REAL WORLD” SYSTEM
A numerical value assigned to each unit of the population
PROBABILTY MODEL
Temp
Mass
Foot length
Shoe size
# children
Coin toss
Pregnant?
Continuous
Continuous
Quantitative
Discrete
Discrete
Qualitative
(Heads = 1, Tails = 0)
Binary
(Yes = 1, No = 0)

5. POPULATION (of “units”)
uniform
skew (positive)
symmetric unimodal
“Arbitrarily large” or infinite collection of units (rocks, toasters, people, districts,…)
X = “Random Variable”
“REAL WORLD” SYSTEM
A numerical value assigned to each unit of the population
PROBABILTY MODEL
Temp
Mass
Foot length
Shoe size
# children
Continuous
Continuous
Quantitative
Discrete
Discrete
“probability density function”
“histogram”
“probability mass function”

6. ~ Three Ingredients of a “Probability Model” ~
Sample space (of an experiment) = set of all possible outcomes
Sigma-field of events
Probability measure
Def: A “random variable” X is a function that maps Ω to the real numbers. | |
Example: Roll one die.
Sample space
{1,2,3,4,5,6}
(Discrete)
Random Variable X = “Value shown”
“probability mass function”
pmf

7. ~ Three Ingredients of a “Probability Model” ~
Sample space (of an experiment) = set of all possible outcomes
Sigma-field of events
Probability measure
Def: A “random variable” X is a function that maps Ω to the real numbers. | |
“Uniform Distribution” X
Example: Roll one die.
fair die.
Sample space
{1,2,3,4,5,6}
(Discrete)
Random Variable X = “Value shown” x
p(x) 1
1/6 2 3 4 5 6
“probability mass function”
Total Area = 1
pmf

8. ~ Three Ingredients of a “Probability Model” ~
Sample space (of an experiment) = set of all possible outcomes
Sigma-field of events
Probability measure
Def: A “random variable” X is a function that maps Ω to the real numbers. | |
{1,2,3,4,5,6}
“Uniform Distribution” X
Example: Roll one die.
fair die.
Sample space
Exercise
(Discrete)
Random Variable X = “Value shown” x
p(x)
0.20
0.30
0.15
0.10
0.05 1 x
p(x) 1
1/6 2 3 4 5 6
“probability mass function”
Total Area = 1
pmf

9. ~ Three Ingredients of a “Probability Model” ~
Sample space (of an experiment) = set of all possible outcomes
Sigma-field of events
Probability measure
Def: A “random variable” X is a function that maps Ω to the real numbers. | |
Example: Roll two fair dice.
Sample space
The set of all ordered pairs (i, j) such that i = 1,2,3,4,5,6 and j = 1,2,3,4,5,6
Die 1
Die 2

10. ~ Three Ingredients of a “Probability Model” ~
Sample space (of an experiment) = set of all possible outcomes
Sigma-field of events
Probability measure
Def: A “random variable” X is a function that maps Ω to the real numbers. | |
Example: Roll two fair dice.
Sample space
The set of all ordered pairs (i, j) such that i = 1,2,3,4,5,6 and j = 1,2,3,4,5,6
Die 1
Die 2
Events
A = “Die1 > Die2”
B = “Roll doubles”
C = “The sum of the two dice is 6.”

11. Def: A “random variable” X is a function that maps Ω to the real numbers.
Example: Roll two fair dice.
Sample space
Die 1
Die 2
Events
A = “Die1 > Die2”
= {(2,1), (3,1), (3,2), (4,1), (4,2),
(4,3), (5,1), (5,2), (5,3), (5,4),
(6,1), (6,2), (6,3), (6,4), (6,5)}
A = “Y > 0”
B = “Roll doubles”
= “Die1 = Die2”
= {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}
B = “Y = 0”

12. Example: Roll two fair dice. Sample space
Def: A “random variable” X is a function that maps Ω to the real numbers.
Example: Roll two fair dice.
Sample space
Die 1
Die 2
Events
C = “The sum of the two dice is 6.”
= “X = 6”
= {(1,5), (2,4), (3,3), (4,2), (5,1)} x
pmf p(x) = P(X = x) 2 3 4 5 6 7 8 9 10 11 12 1
Probability…

13. Example: Roll two fair dice. Sample space
Def: A “random variable” X is a function that maps Ω to the real numbers.
Example: Roll two fair dice.
Sample space
Die 1
Die 2
Events
C = “The sum of the two dice is 6.”
= “X = 6”
= {(1,5), (2,4), (3,3), (4,2), (5,1)} x
Outcomes
pmf p(x) = P(X = x) 2 3 4 5 6 7 8 9 10 11 12 1
Probability…

14. Example: Roll two fair dice. Sample space
Def: A “random variable” X is a function that maps Ω to the real numbers.
Example: Roll two fair dice.
Sample space
Die 1
Die 2
Events
C = “The sum of the two dice is 6.”
= “X = 6”
= {(1,5), (2,4), (3,3), (4,2), (5,1)} x
Outcomes
pmf p(x) = P(X = x) 2
(1,1) 3 4 5 6 7 8 9 10 11 12 1
Probability…

15. Example: Roll two fair dice. Sample space
Def: A “random variable” X is a function that maps Ω to the real numbers.
Example: Roll two fair dice.
Sample space
Die 1
Die 2
Events
C = “The sum of the two dice is 6.”
= “X = 6”
= {(1,5), (2,4), (3,3), (4,2), (5,1)} x
Outcomes
pmf p(x) = P(X = x) 2
(1,1) 3
(1,2), (2,1) 4 5 6 7 8 9 10 11 12 1
Probability…

16. Example: Roll two fair dice. Sample space
Def: A “random variable” X is a function that maps Ω to the real numbers.
Example: Roll two fair dice.
Sample space
Die 1
Die 2
Events
C = “The sum of the two dice is 6.”
= “X = 6”
= {(1,5), (2,4), (3,3), (4,2), (5,1)} x
Outcomes
pmf p(x) = P(X = x) 2
(1,1) 3
(1,2), (2,1) 4
(1,3), (2,2), (3,1) 5 6 7 8 9 10 11 12 1
Probability…

17. Example: Roll two fair dice. Sample space
Def: A “random variable” X is a function that maps Ω to the real numbers.
Example: Roll two fair dice.
Sample space
Die 1
Die 2
Events
C = “The sum of the two dice is 6.”
= “X = 6”
= {(1,5), (2,4), (3,3), (4,2), (5,1)} x
Outcomes
pmf p(x) = P(X = x) 2
(1,1) 3
(1,2), (2,1) 4
(1,3), (2,2), (3,1) 5
(1,4), (2,3), (3,2), (4,1) 6
(1,5), (2,4), (3,3), (4,2), (5,1) 7
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1) 8
(2,6), (3,5), (4,4), (5,3), (6,2) 9
(3,6), (4,5), (5,4), (6,3) 10
(4,6), (5,5), (6,4) 11
(5,6), (6,5) 12
(6,6) 1
Probability…

18. Example: Roll two fair dice. Sample space
Def: A “random variable” X is a function that maps Ω to the real numbers.
Example: Roll two fair dice.
Sample space
Die 1
Die 2
Events
C = “The sum of the two dice is 6.”
= “X = 6”
= {(1,5), (2,4), (3,3), (4,2), (5,1)} x
Outcomes
pmf p(x) = P(X = x) 2
(1,1)
1/36 3
(1,2), (2,1)
2/36 4
(1,3), (2,2), (3,1)
3/36 5
(1,4), (2,3), (3,2), (4,1)
4/36 6
(1,5), (2,4), (3,3), (4,2), (5,1)
5/36 7
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
6/36 8
(2,6), (3,5), (4,4), (5,3), (6,2) 9
(3,6), (4,5), (5,4), (6,3) 10
(4,6), (5,5), (6,4) 11
(5,6), (6,5) 12
(6,6) 1
Probability…

19. Def: A “random variable” X is a function that maps Ω to the real numbers.
Example: Roll two fair dice.
Sample space
Die 1
Die 2
Events
C = “The sum of the two dice is 6.”
= “X = 6”
= {(1,5), (2,4), (3,3), (4,2), (5,1)} x
pmf p(x) = P(X = x) 2
1/36 3
2/36 4
3/36 5
4/36 6
5/36 7
6/36 8 9 10 11 12 1
Probability…
??????
Chapter 2…
Total Area = 1