Chapter 6

Question 1

The joint distribution of two r.v.s X and Y provides . . .

Answer 1

complete information about the probability of the vector (X, Y) falling into any subset of the plane

Question 2

The marginal distribution of X is the. . .

Answer 2

individual distribution of X, “ignoring” the value of Y

Question 3

The conditional distribution of X given
Y = y is the. . .

Answer 3

“updated” distribution for X after observing Y = y

Question 4

The joint CDF of r.v.s X and Y is the function FX,Y given by . . .

Answer 4

Question 5

The joint PMF of discrete r.v.s X and Y is the function pX,Y given
by . . .

Answer 5

Question 6

For discrete r.v.s X and Y, the marginal PMF of X is . . .

Answer 6

Question 7

The operation of summing over the possible values of Y in order to convert the joint PMF into the marginal PMF of X is known as _______________________

Answer 7

marginalizing out Y

Question 8

For discrete r.v.s X and Y, the conditional PMF of Y given X = x is . . .

Answer 8

Question 9

We can also relate the conditional distribution of Y given X = x
to that of X given Y = y, using Bayes’ rule such that . . .

Answer 9

Question 10

Relate the conditional distribution of Y given X = x to that of X given Y = y using the LOTP

Answer 10

Question 11

Random variables X and Y are independent if for all x and y . . .

Answer 11

Question 12

If X and Y are discrete, this is equivalent to the condition __________________ for all x, y, and it is also equivalent to the condition _____________________ for all x, y such that _______________________

Answer 12

Question 13

Remember that in general, the marginal distributions _________ determine the joint distribution BUT in the special case of ________________, the marginal distributions are all we need in order to specify the joint distribution; we can get the joint PMF by . . .

Answer 13

do not
independence
multiplying the marginal PMFs

Question 14

Formally, in order for X and Y to have a continuous joint distribution, we require that the joint CDF _______________ be . . .

Answer 14

Question 15

For continuous r.v.s X and Y with joint PDF fX,Y , the marginal PDF of X is . . .

Answer 15

Question 16

For continuous r.v.s X and Y with joint PDF fX,Y , the conditional PDF of Y given X = x is . . .

Answer 16

Question 17

we can recover the joint PDF fX,Y if we have the conditional PDF fY|X and the corresponding marginal fX :

Answer 17

Question 18

For continuous r.v.s X and Y , we have the following continuous
form of Bayes’ rule:

Answer 18

Question 19

For continuous r.v.s X and Y , we have the following continuous
form of LOTP:

Answer 19

Question 20

How is the independence of continuous r.v.s X and Y determined?

Answer 20

Question 21

Let g be a function from R2 to R. If X and Y are discrete, then E(g(X,Y)) = _______. If X and Y are continuous, then E(g(X,Y)) = _______.

Answer 21

Question 22

Let X and Y be continuous r.v.s (the analogous method also works in the discrete case). Then E(X + Y) = ________________

Answer 22

Question 23

The covariance between r.v.s X and Y is _________________ or equivalently __________________________

Answer 23

Question 24

If X and Y are independent, then their covariance is _________. We say that these r.v.s are ______________

Answer 24

zero
uncorrelated

Question 25

Cov(X, X) = _________

Answer 25

Var(X)

Question 26

Cov(X, Y) = _____________

Answer 26

Cov(Y, X)

Question 27

Cov(X, c) = ____________

Answer 27

0 for any constant c

Question 28

Cov(aX, Y) = _____________

Answer 28

aCov(X, Y) for any constant a

Question 29

Cov(X + Y, Z) = _______________

Answer 29

Cov(X, Z) + Cov(Y, Z)

Question 30

Cov(X + Y, Z + W) = . . .

Answer 30

Cov(X, Z) + Cov(X, W) + Cov(Y, Z) + Cov(Y, W)

Question 31

Var(X + Y) = ____________

Answer 31

Var(X) + Var(Y) + 2Cov(X, Y)

Question 32

The **correlation** between r.v.s X and Y is . . .

Answer 32