Intro to multivariate probability Models

Question 1

What does it mean to have a multivariate probability model

Answer 1

Model involves more than one variable

Question 2

Define random vector

Answer 2

An n-dimensional random vector is a function from a sample space S into R^n

Question 3

If there is no value where X and Y can occur together we assume the joint pmf takes what value?

Answer 3

0

Question 4

Can we obtain The joint pmf from the marginal pmfs

Answer 4

No

Question 5

Can we obtain the marginal pmfs from the joint pmfs

Answer 5

Yes

Question 6

If X and Y are independent what is the mgf of their sum

Answer 6

The product of mgf of X and product of mgf of Y

Question 7

If there is no independence between variables how can we describe their relationship. Name two measures

Answer 7

Strong or weak. Two measures as covariance and correlation

Question 8

Define covariance

Answer 8

Number denoted cov(X,Y) taking values in (minus infinity, infinity)

Question 9

What is Cov(X,X)

Answer 9

Var(X)

Question 10

Why is the sign of covariance important and what is the drawback of the measure of covariance

Answer 10

The sign of the covariance gives information on whether X and Y are
moving in the same (or opposite) direction, however the value itself is not
informative about the strength of the relation

Question 11

What is the correlation

Answer 11

Number describing the linear relationship between X and Y always between -1 and 1. It can only capture Linear dependence

Question 12

If Correlation is 1 what does this mean

Answer 12

There exists α > 0 and β ∈ R such that Y = αX + β.

Question 13

If correlation is -1 what does this mean

Answer 13

There exists α < 0 and β ∈ R such that Y = αX + β.

Question 14

If X and Y are independent RVs what is their correlation and covariance

Answer 14

cov (X , Y ) = ρXY = 0

Question 15

If the covariance or correlation of two RVs is 0 can we say they are independent

Answer 15

NO two dependent
variables can still have null covariance. Also two dependent variables can have 0 correlation their relationship may just not be linear

Question 16

Is cov(x,y)=cov(y,x)

Answer 16

YES

Question 17

What is the marginal distribution of X from a bivariate normal distribution f(x,y)

Answer 17

The marginal distribution of X is N(μX , σ^2X )

Question 18

What is the marginal distribution of Y from a bivariate normal distribution f(x,y)

Answer 18

The marginal distribution of X is N(μY , σ^2XY)

Question 19

How does the cauchy schwarz inequality relate to the holders inequality

Answer 19

Hölder’s Inequality of p = q = 2

is called Cauchy-Schwarz Inequality

Question 20

What inequality can be used to prove correlation<= 1

Answer 20

cov (X , Y ))^2 ≤ σ^2X σ^2Y

Question 21

Define in words a convex function

Answer 21

Convex functions lie below lines connecting any two points.

Convex functions lie above all of its tangents

Question 22

How to tell if a function if concave or convex

Answer 22

g (x) is convex if g′′(x) ≥ 0, for all x, and g (x) is concave if g’′(x) ≤ 0,
for all x

Question 23

State Jensens inequality

Answer 23

For any random variable X , if g (x) is a convex function, then:
E [g (X )] ≥ g (E [X ])

Question 24

Since g (x) = 1/x is convex, what insight can jensens inequality give us

Answer 24

E [1/X ] ≥ 1/E [X ], if x is positive

Question 25

Since g (x) = log(X ) is concave, what insight can jensen’s inequality give us

Answer 25

E [log(X )] ≤ log (E [X ])

Question 26

How can we use jensen’s inequality when a function is concave

Answer 26

Reverse the inequality

E [g (X )] ≤g (E [X ])