A Joint Distribution of A Joint Distribution of Discrete Random VariablesDiscrete Random Variables

X

1 2 3

1 0.15 0.2 0.15 0.5 Marginal

Y 2 0.07 0.14 0.09 0.3 Distribution

3 0.05 0.05 0.1 0.2 Of Y

0.27 0.39 0.34

Marginal Distribution of X

Bivariate U((0,1),(0,1))Bivariate U((0,1),(0,1))

Bivariate Normal Bivariate Normal DistributionDistribution

Three Senators’ Article I Three Senators’ Article I impeachment votes:impeachment votes:

Eight possible outcomesEight possible outcomesLott Wellstone Lieberman

1 1 1

1 1 0

1 0 1

1 0 0

0 0 0

0 1 1

0 1 0

0 0 1

Prob IF p=1/4

p3 1/64

p2(1-p) 3/64

p2(1-p) 3/64

p(1-p)2 9/64

(1-p)3 27/64

p2(1-p) 3/64

p(1-p)2 9/64

p(1-p)2 9/64

Observed

Bernoulli Likelihood Bernoulli Likelihood Function, Three Function, Three observation caseobservation case

The Effect of The Effect of k(y)k(y)

Two ExtremaTwo Extrema

The MLEThe MLE

MLE

Bernoulli Likelihood and Bernoulli Likelihood and Log-likelihood, three Log-likelihood, three

observation caseobservation case

Ln(exports) by country, Ln(exports) by country, 20012001

Frac

tion

Ln(Exports)2.07944 4.76217

0

.2