Final Exam [Bulk Review 2]

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

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

Answer 7

Question 8

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

Answer 8

Question 9

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

Answer 9

Question 10

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

Answer 10

Question 11

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

Answer 11

zero
uncorrelated

Question 12

Cov(X, X) = _________

Answer 12

Var(X)

Question 13

Cov(X, Y) _____ Cov(Y, X)

Answer 13

=

Question 14

Cov(X, c) = ____________

Answer 14

0 for any constant c

Question 15

Cov(aX, Y) = _____________

Answer 15

aCov(X, Y) for any constant a

Question 16

Cov(X + Y, Z) = _______________

Answer 16

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

Question 17

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

Answer 17

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

Question 18

Var(X + Y) = ____________

Answer 18

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

Question 19

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

Answer 19

Question 20

We say that c is a median of a random variable X if _______________

Answer 20

P(X ≤ c) ≥ 1/2 and P(X ≥ c) ≥ 1/2

Question 21

For a discrete r.v. X, we say that c is a mode of X if it maximizes the PMF: _____________. For a continuous r.v. X with PDF f, we say that c is a mode if it maximizes the PDF: _______________

Answer 21

P(X = c) ≥ P(X = x) for all x

f (c) ≥ f (x) for all x

Question 22

Measures of central tendency are . . .
Measures of spread are . . .
Skewness measures . . .

Answer 22

mean, median, mode
standard deviation, variance
asymmetry

Question 23

Describe the 3 types of skewness

Answer 23

negative/left
not skewed
positive/right

Question 24

Let X be an r.v. with mean μ and variance σ2. For any positive integer n, the nth moment of X is __________, the nth central moment is ___________, and the nth standardized moment and the nth standardized moment in . .

Answer 24

E(Xn)
E((X − μ)n)

Question 25

The skewness of an r.v. X with mean μ and variance σ2 is the _________ standardized moment of X:

Answer 25

third

Question 26

Let X be symmetric about its mean μ. Then for any odd number m, the mth central moment E(X − μ)m is ________

Answer 26

0 if it exists

Question 27

The moment generating function (MGF) of an r.v. X is . . . Otherwise we say the MGF of X ___________

Answer 27

M(t) = E(e^tX ), as a function of t, if this is finite on some open interval (−a, a) containing 0

does not exist

Question 28

For X ∼ Bern(p), e^tX takes on the value et with probability p and the value 1 with probability q, so M(t) = ____________ Since
this is finite for all values of t, the MGF is defined _____________

Answer 28

E(e^ tX ) = pe^t + q
on the entire real line

Question 29

For X ∼ Geom(p), M(t) = E(e^tX ) = _____

Answer 29

Question 30

Let U ∼ Unif (a, b). Then the MGF of U is . . .

Answer 30

Question 31

The MGF of X ∼ Expo(λ) when t < λ is . . .

Answer 31

Question 32

Given the MGF of X, we can get the nth moment of X by evaluating the nth ____________ of the MGF at 0:

The Taylor expansion of M(t) about 0 is:

Answer 32

derivative

Question 33

The MGF of a random variable determines its distribution meaning that . . .

Answer 33

If two r.v.s have the same MGF, they must have the same distribution. In fact, if there is even a tiny interval (−a, a) containing 0 on which the MGFs are equal, then the r.v.s must have the same distribution.

Question 34

If X and Y are independent, then the MGF of X + Y is the _________ of the individual MGFs

Answer 34

product

Question 35

Give the MGF of the standard normal and normal distributions

Answer 35

Question 36

Give the MGF of the exponential distribution

Answer 36

Question 37

A ______________ on a probability gives a provable guarantee that the probability is in a certain range.
These inequalities will often allow us to narrow down the range of possible values for the exact answer, that is, to . . .

Answer 37

bound
determine an upper bound and/or lower bound

Question 38

Let the number of items produced in a factory during a week be a continuous random variable with E(X) = 50. Give
an upper bound for P(X > 75).

Answer 38

Question 39

Explain the weak law of large numbers.

Answer 39

Question 40

Explain the central limit theorem and its use in approximation.

Answer 40