Lecture 7: Random Variables and Probability …I Introduce probability distributions, which are...

Lecture 7:Random Variables and Probability Distributions

API-201Z

Maya Sen

Harvard Kennedy Schoolhttp://scholar.harvard.edu/msen

Announcements

I Problem Set #4 posted

I We’ll be posting several practice exams and practice problemsin advance of midterm

Announcements

Roadmap

I By now, be comfortable w/ summary statistics in Stata/R,basic probability, conditional probability, independence, LOTP,Bayes’ Rule

I Now:I Introduce probability distributions, which are foundational for

statistical inferenceI Random variablesI Give examples of discrete and continuous random variablesI Walk through probability distributions for discrete random

variables (continuous next time)I Introduce Bernoulli processes

Roadmap

I Now:

I Introduce probability distributions, which are foundational forstatistical inference

I Random variablesI Give examples of discrete and continuous random variablesI Walk through probability distributions for discrete random

Roadmap

statistical inference

I Random variablesI Give examples of discrete and continuous random variablesI Walk through probability distributions for discrete random

Roadmap

statistical inferenceI Random variables

I Give examples of discrete and continuous random variablesI Walk through probability distributions for discrete random

Roadmap

statistical inferenceI Random variablesI Give examples of discrete and continuous random variables

I Walk through probability distributions for discrete randomvariables (continuous next time)

I Introduce Bernoulli processes

Roadmap

variables (continuous next time)

I Introduce Bernoulli processes

Roadmap

What are random variables?

I Random phenomenon: A situation where we know whatoutcomes could happen, but we don’t know which particularoutcome did or will happen

I Ex) Flipping a coin 3 timesI Ex) Randomly calling 5 U.S. SenatorsI Ex) Lady Bristol choosing 4 cups of tea out of 8

I Random variables: (1) Numerical outcomes of (2) randomphenomenon

I Ex) Number of Heads in 3 coin flipsI Ex) Number of female Democrats you callI Ex) Number of 4 cups Lady identifies correctlyI Can take on a set of possible different numerical values, each

with an associated probabilityI RVs denoted by capital letters – e.g., X or YI Note: Random variable = number (e.g, number of heads),

random event = event (e.g., tossing heads)

I As before, “random” does not mean “haphazard”

I Random phenomenon

: A situation where we know whatoutcomes could happen, but we don’t know which particularoutcome did or will happen

I Ex) Flipping a coin 3 times

I Ex) Randomly calling 5 U.S. SenatorsI Ex) Lady Bristol choosing 4 cups of tea out of 8

I Ex) Flipping a coin 3 timesI Ex) Randomly calling 5 U.S. Senators

I Ex) Lady Bristol choosing 4 cups of tea out of 8

I Random variables

: (1) Numerical outcomes of (2) randomphenomenon

I Ex) Number of Heads in 3 coin flips

I Ex) Number of female Democrats you callI Ex) Number of 4 cups Lady identifies correctlyI Can take on a set of possible different numerical values, each

I Ex) Number of Heads in 3 coin flipsI Ex) Number of female Democrats you call

I Ex) Number of 4 cups Lady identifies correctlyI Can take on a set of possible different numerical values, each

I Ex) Number of Heads in 3 coin flipsI Ex) Number of female Democrats you callI Ex) Number of 4 cups Lady identifies correctly

I Can take on a set of possible different numerical values, eachwith an associated probability

I RVs denoted by capital letters – e.g., X or YI Note: Random variable = number (e.g, number of heads),

with an associated probability

I RVs denoted by capital letters – e.g., X or YI Note: Random variable = number (e.g, number of heads),

with an associated probabilityI RVs denoted by capital letters – e.g., X or Y

I Note: Random variable = number (e.g, number of heads),random event = event (e.g., tossing heads)

I Example: You toss a coin 3 times

I Random phenomenon → act of tossing the coin 3 timesI Random variable →

I Sequence of outcomes of the flips, e.g. HTH, is not a randomvariable (not a number)

I But we could make it into a random variable, X , by making it# of Heads in 3 flips

I Note: Sample space can have many different random variablesdefined on it (e.g., number of tails flips, Y )

I Random variable → one of several possible mapping fromsample space (HTH) into a number (in this case, 2)

I Random phenomenon → act of tossing the coin 3 times

I Random variable →I Sequence of outcomes of the flips, e.g. HTH, is not a random

variable (not a number)I But we could make it into a random variable, X , by making it

# of Heads in 3 flips

Discrete Random Variables

I A discrete random variable can take on only integer(countable) number of values (usually within an interval)

I Examples:I Outcomes when you roll a roulette wheelI Number of times a particular word is used in a documentI Number of Democrats in U.S. Senate

I → In theory, you could count up these values, given finiteamount of time

I Examples:

I Outcomes when you roll a roulette wheelI Number of times a particular word is used in a documentI Number of Democrats in U.S. Senate

I Examples:I Outcomes when you roll a roulette wheel

I Number of times a particular word is used in a documentI Number of Democrats in U.S. Senate

I Examples:I Outcomes when you roll a roulette wheelI Number of times a particular word is used in a document

I Number of Democrats in U.S. Senate

Continuous Random Variables (next time)

I A continuous random variable that can take on any value(usually within an interval) on the real number line

I Examples:I Annual rainfall in BangladeshI Time it takes to run a 100m dashI Time until next earthquake in Japan

I → Cannot count these quantities, given finite amount of time

I Much of intuition we cover today applies to both

I Examples:

I Annual rainfall in BangladeshI Time it takes to run a 100m dashI Time until next earthquake in Japan

I Examples:I Annual rainfall in Bangladesh

I Time it takes to run a 100m dashI Time until next earthquake in Japan

I Examples:I Annual rainfall in BangladeshI Time it takes to run a 100m dash

I Time until next earthquake in Japan

How do we describe random variables?

I We describe distribution of random variables using probabilitydistributions

I Probability distributions represent information about howlikely various outcomes of X are

I 2 common ways of representing probability distributions fordiscrete random variables are via:

1. Probability mass functions (f (x))2. Cumulative mass functions (F (x))

I Similar for continuous distributions: probability densityfunctions, cumulative density functions

1. Probability mass functions (f (x))

2. Cumulative mass functions (F (x))

Probability mass functions (PMFs)

I Probability mass functions: A function that defines theprobability of each possible outcome

f (x) = P(X = x)

where X is the random variable, x possible value it could take

I Provides you with all possible outcomes and probability ofeach outcome

I Three ways of describing PMFsI TableI Relative frequency histogram (bar plot)I Formula itself

f (x) = P(X = x)

I Three ways of describing PMFs

I TableI Relative frequency histogram (bar plot)I Formula itself

f (x) = P(X = x)

I Three ways of describing PMFsI Table

I Relative frequency histogram (bar plot)I Formula itself

f (x) = P(X = x)

I Three ways of describing PMFsI TableI Relative frequency histogram (bar plot)

I Formula itself

f (x) = P(X = x)

Discrete Example

I Let X be the number of heads in 3 coin flips

I Sample space of possible outcomes would look like:

Event Value of X

1) TTT 02) TTH 13) THT 14) HTT 15) THH 26) HTH 27) HHT 28) HHH 3

Discrete Example

Event Value of X

Discrete Example

Event Value of X

Discrete Example

Event Value of X

Discrete Example

Event Value of X

1) TTT 0

2) TTH 13) THT 14) HTT 15) THH 26) HTH 27) HHT 28) HHH 3

Discrete Example

Event Value of X

Discrete Example

I Probability mass function distribution would look like

0 1 2 3

f (x) 1/8 3/8 3/8 1/8

I All probabilities, between 0 and 1, inclusive

I Probabilities add up to 1

I Note: You’d only know this distribution if you repeated therandom phenomenon over and over again!

Discrete Example

0 1 2 3

f (x) 1/8 3/8 3/8 1/8

Discrete Example

0 1 2 3

f (x) 1/8 3/8 3/8 1/8

Discrete Example

0 1 2 3

f (x) 1/8 3/8 3/8 1/8

Discrete Example

0 1 2 3

f (x) 1/8 3/8 3/8 1/8

Discrete Example

0 1 2 3

f (x) 1/8 3/8 3/8 1/8

Discrete Example

0 1 2 3

f (x) 1/8 3/8 3/8 1/8

Discrete Example

0 1 2 3

f (x) 1/8 3/8 3/8 1/8

Discrete Example

I Could also represent this graphically:

X = 0 X = 1 X = 2 X = 3

Probability Mass Distribution

I Again note: Heights of bars add up to 1

Discrete Example

X = 0 X = 1 X = 2 X = 3

Discrete Example

X = 0 X = 1 X = 2 X = 3

Discrete Example

X = 0 X = 1 X = 2 X = 3

Cumulative Mass Function

I Probability mass function describes the probability of eachpossible outcome, f (x) = P(X = x)

I A cumulative mass function (CMF) gives us the probabilitythat a random variable X takes on a value less than or equalto some particular value x :

F (x) = P(X 6 x)

=∑X6x

P(X = x)

I Can also represent CMFs using:I TableI Relative frequency histogramI Formula itself

F (x) = P(X 6 x)

=∑X6x

P(X = x)

F (x) = P(X 6 x)

=∑X6x

P(X = x)

F (x) = P(X 6 x)

=∑X6x

P(X = x)

F (x) = P(X 6 x)

=∑X6x

P(X = x)

I Can also represent CMFs using:

I TableI Relative frequency histogramI Formula itself

F (x) = P(X 6 x)

=∑X6x

P(X = x)

I Can also represent CMFs using:I Table

I Relative frequency histogramI Formula itself

F (x) = P(X 6 x)

=∑X6x

P(X = x)

I Can also represent CMFs using:I TableI Relative frequency histogram

I Formula itself

F (x) = P(X 6 x)

=∑X6x

P(X = x)

Discrete Example

I Universe of possible outcomes would look like:

Event Value of X

Discrete Example

Event Value of X

Discrete Example

Event Value of X

Discrete Example

Event Value of X

Discrete Example

0 1 2 3

f (x) 1/8 3/8 3/8 1/8

I Cumulative mass function distribution would look like:

0 1 2 3

F (x) 1/8 4/8 7/8 1

Discrete Example

0 1 2 3

f (x) 1/8 3/8 3/8 1/8

0 1 2 3

F (x) 1/8 4/8 7/8 1

Discrete Example

0 1 2 3

f (x) 1/8 3/8 3/8 1/8

0 1 2 3

F (x) 1/8 4/8 7/8 1

Discrete Example

0 1 2 3

f (x) 1/8 3/8 3/8 1/8

0 1 2 3

F (x) 1/8 4/8 7/8 1

Discrete Example

0 1 2 3

f (x) 1/8 3/8 3/8 1/8

0 1 2 3

F (x) 1/8 4/8 7/8 1

Discrete Example

0 1 2 3

f (x) 1/8 3/8 3/8 1/8

0 1 2 3

F (x) 1/8 4/8 7/8 1

Discrete Example

I And then we can also represent the CMF graphically:

X = 0 X = 1 X = 2 X = 3

X ≤ 0 X ≤ 1 X ≤ 2 X ≤ 3

Cumulative Mass Distribution

Discrete Example

X = 0 X = 1 X = 2 X = 3

X ≤ 0 X ≤ 1 X ≤ 2 X ≤ 3

Discrete Example

X = 0 X = 1 X = 2 X = 3

X ≤ 0 X ≤ 1 X ≤ 2 X ≤ 3

Sex Discrimination Example

I There are 5 candidates for 2 job openingsI 3 of the candidates are women and 2 are men:

I W1, W2, W3, M1, M2

I Let X be a random variable that is the # of women hiredI X ∈ {0, 1, 2}

I What is the PMF of X?

I What is the CMF of X?

I There are 5 candidates for 2 job openings

I 3 of the candidates are women and 2 are men:I W1, W2, W3, M1, M2

I W1, W2, W3, M1, M2

I Let X be a random variable that is the # of women hired

I X ∈ {0, 1, 2}

I W1, W2, W3, M1, M2

10 possibilities in the sample space:

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

I Probability mass function distribution:

X = 0 X = 1 X = 2

f (x) ? ? ?

I Cumulative mass function distribution:

X 6 0 X 6 1 X 6 2

F (x) ? ? ?

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

X = 0 X = 1 X = 2

f (x) ? ? ?

X 6 0 X 6 1 X 6 2

F (x) ? ? ?

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

X = 0 X = 1 X = 2

f (x) ? ? ?

X 6 0 X 6 1 X 6 2

F (x) ? ? ?

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

X = 0 X = 1 X = 2

f (x) ? ? ?

X 6 0 X 6 1 X 6 2

F (x) ? ? ?

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

X = 0 X = 1 X = 2

f (x) ? ? ?

X 6 0 X 6 1 X 6 2

F (x) ? ? ?

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

X = 0 X = 1 X = 2

f (x) ? ? ?

X 6 0 X 6 1 X 6 2

F (x) ? ? ?

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

X = 0 X = 1 X = 2

f (x) ? ? ?

X 6 0 X 6 1 X 6 2

F (x) ? ? ?

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

X = 0 X = 1 X = 2

f (x) 1/10 ? ?

X 6 0 X 6 1 X 6 2

F (x) ? ? ?

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

X = 0 X = 1 X = 2

f (x) 1/10 ? ?

X 6 0 X 6 1 X 6 2

F (x) ? ? ?

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

X = 0 X = 1 X = 2

f (x) 1/10 6/10 ?

X 6 0 X 6 1 X 6 2

F (x) ? ? ?

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

X = 0 X = 1 X = 2

f (x) 1/10 6/10 ?

X 6 0 X 6 1 X 6 2

F (x) ? ? ?

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

X 6 0 X 6 1 X 6 2

F (x) ? ? ?

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

X 6 0 X 6 1 X 6 2

F (x) 1/10 ? ?

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

X 6 0 X 6 1 X 6 2

F (x) 1/10 7/10 ?

I (W1, W2) (W1, W3) (W1, M1) (W1, M2)

I (W2, W3) (W2, M1) (W2, M2)

I (W3, M1) (W3, M2)

I (M1, M2)

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

X 6 0 X 6 1 X 6 2

F (x) 1/10 7/10 1

I Can use this information to address substantive questions

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

X 6 0 X 6 1 X 6 2

F (x) 1/10 7/10 1

I Would you be concerned if 1 or fewer women were hired?

I Would you be concerned if exactly 1 woman was hired?

I Would you be concerned if no women were hired?

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

X 6 0 X 6 1 X 6 2

F (x) 1/10 7/10 1

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

X 6 0 X 6 1 X 6 2

F (x) 1/10 7/10 1

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

X 6 0 X 6 1 X 6 2

F (x) 1/10 7/10 1

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

X 6 0 X 6 1 X 6 2

F (x) 1/10 7/10 1

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

X 6 0 X 6 1 X 6 2

F (x) 1/10 7/10 1

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

X 6 0 X 6 1 X 6 2

F (x) 1/10 7/10 1

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

X 6 0 X 6 1 X 6 2

F (x) 1/10 7/10 1

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

X 6 0 X 6 1 X 6 2

F (x) 1/10 7/10 1

Key features of probability distributions

I Similar to a sample of data, we can calculate useful info aboutprobability distribution:

I Expected valueI VarianceI Standard deviation

I Use Greek letters to denote these population parameters

I Can’t actually measure these empirically, but can calculatethem based on what we think would happen if we repeatedthe random process over and over again

I Conceptually distinct from (but completely analogous to)sample parameters from earlier

I Expected value

I VarianceI Standard deviation

I Expected valueI Variance

I Standard deviation

Expected value

I Expected value: a measure of the center of a probabilitydistribution

I Denoted as µ

E [X ] = µX

=∑all x

I Sum of all possible values, each weighted by its relativeprobability

I Substantively: If we repeated random process repeatedly andaveraged values together, what would we expect to find?

Expected value

I Denoted as µ

E [X ] = µX

=∑all x

Expected value

I Denoted as µ

E [X ] = µX

=∑all x

Expected value

I Denoted as µ

E [X ] = µX

=∑all x

Expected value

I Denoted as µ

E [X ] = µX

=∑all x

Expected value

I Denoted as µ

E [X ] = µX

=∑all x

Variance and Standard Deviation

I Variance: Measure of the spread of a probability distribution:

V [X ] = σ2

=∑all x

(x − µ)2p(x)

I Standard deviation: Square root of the variance

SD[X ] = σ

=√σ2

V [X ] = σ2

=∑all x

(x − µ)2p(x)

SD[X ] = σ

=√σ2

V [X ] = σ2

=∑all x

(x − µ)2p(x)

SD[X ] = σ

=√σ2

V [X ] = σ2

=∑all x

(x − µ)2p(x)

SD[X ] = σ

=√σ2

V [X ] = σ2

=∑all x

(x − µ)2p(x)

SD[X ] = σ

=√σ2

Probability mass function distribution from earlier:

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

10× 0

10× 1

10× 2

(0 −

(1 −

(2 −

Sex Discrimination ExampleProbability mass function distribution from earlier:

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

10× 0

10× 1

10× 2

(0 −

(1 −

(2 −

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

10× 0

10× 1

10× 2

(0 −

(1 −

(2 −

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

10× 0

10× 1

10× 2

(0 −

(1 −

(2 −

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

10× 0

10× 1

10× 2

(0 −

(1 −

(2 −

X = 0 X = 1 X = 2

f (x) 1/10 6/10 3/10

10× 0

10× 1

10× 2

(0 −

(1 −

(2 −

Joint Probability Distributions

I Have focused on distribution of one random variable, X

I A joint probability mass function describes behavior of 2discrete random variables, X and Y , at same time

I Lists probabilities for all pairs of possible values of X and Y

I Can represent as relative frequency table, joint frequencyhistogram, or formula

I Written as P(X = x ,Y = y)

I Can also have conditional probabilities: (X = x |Y = y)

I And can use joint probability P(X = x ,Y = y) to back outmarginal probability distribution, P(X = x) or P(Y = y)

I Ex) Suppose in Boston there is relationship between floods &snowfall in a given year

Association between floods and snowfalls

# of Snowstorms (Y)0 1 2 3 4 Total

# of Floods (X)

0 0.15 0.05 0 0 0 0.21 0.1 0.1 0.05 0 0 0.252 0.05 0.05 0.1 0.05 0 0.253 0 0.05 0.05 0.07 0.03 0.24 0 0 0 0.03 0.07 0.1

Total 0.3 0.25 0.2 0.15 0.1 1.00

# of Floods (X)

0 0.15 0.05 0 0 0 0.21 0.1 0.1 0.05 0 0 0.252 0.05 0.05 0.1 0.05 0 0.253 0 0.05 0.05 0.07 0.03 0.24 0 0 0 0.03 0.07 0.1

Total 0.3 0.25 0.2 0.15 0.1 1.00

I What is probability of 2 floods and 3 snowstorms in arandomly chosen year?

# of Floods (X)

0 0.15 0.05 0 0 0 0.21 0.1 0.1 0.05 0 0 0.252 0.05 0.05 0.1 0.05 0 0.253 0 0.05 0.05 0.07 0.03 0.24 0 0 0 0.03 0.07 0.1

Total 0.3 0.25 0.2 0.15 0.1 1.00

I What is probability of 2 floods and 3 snowstorms in arandomly chosen year?

# of Floods (X)

0 0.15 0.05 0 0 0 0.21 0.1 0.1 0.05 0 0 0.252 0.05 0.05 0.1 0.05 0 0.253 0 0.05 0.05 0.07 0.03 0.24 0 0 0 0.03 0.07 0.1

Total 0.3 0.25 0.2 0.15 0.1 1.00

I What is the marginal probability mass function of Y ?

# of Floods (X)

0 0.15 0.05 0 0 0 0.21 0.1 0.1 0.05 0 0 0.252 0.05 0.05 0.1 0.05 0 0.253 0 0.05 0.05 0.07 0.03 0.24 0 0 0 0.03 0.07 0.1

Total 0.3 0.25 0.2 0.15 0.1 1.00

I What is the marginal probability mass function of Y ?

# of Floods (X)

0 0.15 0.05 0 0 0 0.21 0.1 0.1 0.05 0 0 0.252 0.05 0.05 0.1 0.05 0 0.253 0 0.05 0.05 0.07 0.03 0.24 0 0 0 0.03 0.07 0.1

Total 0.3 0.25 0.2 0.15 0.1 1.00

I What is the conditional probability mass function of Yconditional on X = 2?

# of Floods (X)

0 0.15 0.05 0 0 0 0.21 0.1 0.1 0.05 0 0 0.252 0.05 0.05 0.1 0.05 0 0.253 0 0.05 0.05 0.07 0.03 0.24 0 0 0 0.03 0.07 0.1

Total 0.3 0.25 0.2 0.15 0.1 1.00

# of Floods (X)

0 0.15 0.05 0 0 0 0.21 0.1 0.1 0.05 0 0 0.252 0.20 0.20 0.40 0.20 0 1.003 0 0.05 0.05 0.07 0.03 0.24 0 0 0 0.03 0.07 0.1

Total 0.3 0.25 0.2 0.15 0.1 1.00

Probability Distribution Families

I We can write down some PMFs and CMFs for easy examples→ Real world much more complicated

I Fortunately, don’t have to calculate our own PMFs/CMFs forevery random variable we are interested in

I Generations of statisticians have derived generalizable PMFs,CMFs for many frequently occurring processes

I This includes processes frequently seen in social sciences/policyI Also includes processes frequently seen in natural sciences

(e.g., Normal distribution)I → Wikipedia a great resource

I This includes processes frequently seen in social sciences/policy

I Also includes processes frequently seen in natural sciences(e.g., Normal distribution)

I → Wikipedia a great resource

(e.g., Normal distribution)

I → Wikipedia a great resource

Bernoulli Distribution

I A single coin flip well-known example of one of most basicprobability distributions, a Bernoulli distribution/trial

I Need:

I A trial (e.g., coin flip) that has a probability of success (p)and a probability of failure (1 − p)

I For Bernoulli processes, the PMF:

f (x) =

{p for x = 11 − p for x = 0

I which can be rewritten as

f (x) = px(1 − p)1−x

I Need:

f (x) =

{p for x = 11 − p for x = 0

f (x) = px(1 − p)1−x

I Need:

f (x) =

{p for x = 11 − p for x = 0

f (x) = px(1 − p)1−x

I Need:

f (x) =

{p for x = 11 − p for x = 0

f (x) = px(1 − p)1−x

I Need:

f (x) =

{p for x = 11 − p for x = 0

f (x) = px(1 − p)1−x

I Need:

f (x) =

{p for x = 11 − p for x = 0

f (x) = px(1 − p)1−x

Binomial Distribution

I A series of Bernoulli trial results in a binomial distribution

I Ex) Series of coin flipsI Need:

I Only two possible outcomes at each trial (success and failure)I Constant probability of success, p, for each trialI A fixed number of trials n that are independentI X represents the number of success in n trials

I Ex) Series of coin flips

I Need:I Only two possible outcomes at each trial (success and failure)I Constant probability of success, p, for each trialI A fixed number of trials n that are independentI X represents the number of success in n trials

I Only two possible outcomes at each trial (success and failure)

I Constant probability of success, p, for each trialI A fixed number of trials n that are independentI X represents the number of success in n trials

I Only two possible outcomes at each trial (success and failure)I Constant probability of success, p, for each trial

I A fixed number of trials n that are independentI X represents the number of success in n trials

I Only two possible outcomes at each trial (success and failure)I Constant probability of success, p, for each trialI A fixed number of trials n that are independent

I X represents the number of success in n trials

I For a binomial process, denote X ∼ Binomial(n, p)

I Has a PMF of:

P(X = x) =

)px(1 − p)n−x

I where(nx

)is the binomial coefficient: gives us the possible

ways of getting x successes in n trials

x!(n−x)!

I and n! = n × (n − 1)× (n − 2)× ...× (1)

I and a CMF of:

P(X 6 x) =∑(

)px(1 − p)n−x

I Has a PMF of:

P(X = x) =

)px(1 − p)n−x

I where(nx

x!(n−x)!

I and n! = n × (n − 1)× (n − 2)× ...× (1)

I and a CMF of:

P(X 6 x) =∑(

)px(1 − p)n−x

I Has a PMF of:

P(X = x) =

)px(1 − p)n−x

I where(nx

x!(n−x)!

I and n! = n × (n − 1)× (n − 2)× ...× (1)

I and a CMF of:

P(X 6 x) =∑(

)px(1 − p)n−x

I Has a PMF of:

P(X = x) =

)px(1 − p)n−x

I where(nx

x!(n−x)!

I and n! = n × (n − 1)× (n − 2)× ...× (1)

I and a CMF of:

P(X 6 x) =∑(

)px(1 − p)n−x

I Has a PMF of:

P(X = x) =

)px(1 − p)n−x

I where(nx

x!(n−x)!

I and n! = n × (n − 1)× (n − 2)× ...× (1)

I and a CMF of:

P(X 6 x) =∑(

)px(1 − p)n−x

I Has a PMF of:

P(X = x) =

)px(1 − p)n−x

I where(nx

x!(n−x)!

I and n! = n × (n − 1)× (n − 2)× ...× (1)

I and a CMF of:

P(X 6 x) =∑(

)px(1 − p)n−x

I Has a PMF of:

P(X = x) =

)px(1 − p)n−x

I where(nx

x!(n−x)!

I and n! = n × (n − 1)× (n − 2)× ...× (1)

I and a CMF of:

P(X 6 x) =∑(

)px(1 − p)n−x

I Has a PMF of:

P(X = x) =

)px(1 − p)n−x

I where(nx

x!(n−x)!

I and n! = n × (n − 1)× (n − 2)× ...× (1)

I and a CMF of:

P(X 6 x) =∑(

)px(1 − p)n−x

I Has a PMF of:

P(X = x) =

)px(1 − p)n−x

I where(nx

x!(n−x)!

I and n! = n × (n − 1)× (n − 2)× ...× (1)

I and a CMF of:

P(X 6 x) =∑(

)px(1 − p)n−x

Other useful discrete families

I Poisson Distribution: X is # of events per unit of timeI Probability that 25 aviation incidents occur in a given yearI Probability that 5 MBTA buses stop at JFK bus stop in an hr

I Geometric Distribution: X is the # of Bernoulli trials neededto get one “success”

I Probability that HKS needs to ask 5 former ambassadors tojoin faculty before one says “yes”

I Probability that Senate will confirm 10 of President’s judicialappointments before it rejects a single one

I Many preprogrammed in Stata or R

I Wikipedia: Detailed explanations

I Poisson Distribution: X is # of events per unit of time

I Probability that 25 aviation incidents occur in a given yearI Probability that 5 MBTA buses stop at JFK bus stop in an hr

I Poisson Distribution: X is # of events per unit of timeI Probability that 25 aviation incidents occur in a given year

I Probability that 5 MBTA buses stop at JFK bus stop in an hr

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