Econometrics | Ch. 2: Review of Probability

The sample mean is denoted by Y(bar). True or False. Y(bar) is a random variable.

True

E[Y(bar)] = ?

Uy

Explain the law of large numbers (LLN).

Y(bar) will be close to Uy with high probability when the sample is large

For an asymptotic distribution, estimations of Uy by _____ become ___________________ as _________________________.

Y(bar); exact; n approaches infinity

Explain why Y(bar) is itself a random variable.

Y1 and Y2 are chosen at random, where the value of each Yi is random. Thus, the average of them is random.

What are the 3 assumptions we make for the law of large numbers (LLN)?

Yi is i.i.d. variance of Yi is finite outliers are unlikely because var(Yi) is finite

How do you "standardize" a normal distribution? Write the formula.

Z = (Y - Uy) / S.D.y answer on p. 12 of text. notes

What is a probability distribution function (pdf)?

a list of possible values of a variable and the probability that each value will occur

What is a cumulative distribution function (cdf)?

a list of possible values of a variable and the probability that the variable is less than or equal to a particular value

Define random variable (RV).

a numerical summary of a random outcome

What is an event?

a set of one or more outcomes

What is a continuous random variable?

an RV that takes on any possible values of the real number line

What is a discrete random variable?

an RV that takes on whole number values

Write the formula for corr(X,Y) or correlation between X and Y.

answer on p. 11 of text. notes

var(a + bY) = ? Write the formula.

answer on p. 11 of text. notes

var(aX + bY) = ? Write the formula.

answer on p. 11 of text. notes

Write the notation for a standard normal distribution.

answer on p. 12 of text. notes

s.d.[Y(bar)] = ?

answer on p. 13 of text. notes

var[Y(bar)] = ?

answer on p. 13 of text. notes

Write the notation of a normal distribution of RV Y(bar)

answer on p. 14 of text. notes

Write the notation of a normal distribution of RV Y.

answer on p. 14 of text. notes

Write the formula for computing the expectation or mean of RV "Y".

answer on p. 4 of text. notes

Write the formula for computing the first moment of a distribution.

answer on p. 4 of text. notes

Write the formula for computing the second moment of a distribution.

answer on p. 4 of text. notes

Write the formula for computing the variance of RV "Y".

answer on p. 4 of text. notes

Write the formula for computing the third moment of a distribution.

answer on p. 6 of text. notes

Write the formula for computing the fourth moment of a distribution.

answer on p. 7 of text. notes

n observations in simple random sampling are denoted Y1, Y2, ... , Yn. Explain why Y1 and Y2 are are identically and independently distributed (i.i.d.).

because they are randomly drawn, they have the same marginal distribution and are thus, identically distributed because they are drawn independently of one another, they are independently distributed

The mean measures the __________________ of a distribution.

center

X and Y are independent if the ______________________ distribution of Y given X is equal to the _____________________ distribution of Y. Also, write this formula.

conditional; marginal Pr(Y=b | X=a) = Pr(Y=b)

What are the 2 types of random variables?

discrete continuous

If X and Y are independent, then this implies that they are _________________________.

either linearly independent or non-linearly independent OR both

What are the 2 types of sampling distributions?

exact or finite-sample distribution of Y(bar) asymptotic distribution

2 RVs X and Y are ________________________ distributed if or simply ____________________ if knowing the value of one RV provides no information about the other.

independently; independent

The marginal probability of a RV is ________________________.

its probability distribution

The fourth moment of a distribution is the ____________________.

kurtosis

The skewness measures the ________________________ of a distribution.

lack of symmetry

If a distribution's kurtosis is large then it has a __________________ amount of mass in its tails.

large

What are the 2 tools that are used to approximate sampling distributions when the sample size is large?

law of large numbers (LLN) central limit theorem (CLT)

The first moment of a distribution is the ____________________.

mean

What are the standard normal values for the 4 moments of a distribution?

mean -- 0 variance -- 1 skewness -- 0 kurtosis -- 3

Describe simple random sampling.

n objects are selected at random from a population where each object of the population is equally likely to be included into the sample

For an exact distribution, if Y is _______________________________ and Yi is ___________________, then the exact distribution of Y(bar) is ____________________.

normally distributed i.i.d. normal

The third moment of a distribution is the ____________________.

skewness

The standard deviation measures the ____________________ of a distribution.

spread

Pr(Y=b) = ? Write the formula.

sum Pr(X=xi , Y=b) detailed answer on p. 8 of text. notes

If X and Y are dependent, then E(Y | X=a) = ?

sum Yi * Pr(Y=Yi | X=a)

If X and Y are independent, then E(Y | X=a) = ?

sum Yi * Pr(Y=Yi)

A skewness = 0 implies that the distribution is ______________________.

symmetric

The conditional probability of a RV is ________________________.

the distribution of a RV conditional on another RV taking on a certain value

What is the conditional expectation or mean of RV Y? Also, write the expectation notation and the formula.

the expected value of Y, computed using the conditional distribution of Y given X E(Y | x=a) sum yi * Pr(Y=yi | X = a) detailed answer on p. 9 of text. notes

What is covariance? Write the formula for covariance of X and Y.

the extent to which 2 RVs move together answer on p. 11 of text. notes

Define the probability of an outcome.

the proportion of time the outcome occurs in the long -run

Explain the central limit theorem (CLT).

the sampling distribution of the standardized sample average is approximately normal

Define sample space.

the set of all possible outcomes

The kurtosis measures the _________________________ of a distribution.

the weight in the tails

Write the variance notation and the formula for the conditional variance of RV Y.

var(Y | X=a) sum [yi - E(Y | x=a)]^2 * Pr(Y=yi | X=a)

If X and Y are independent, then var(Y | X=a) = ?

var(Yi)

The second moment of a distribution is the ____________________.

variance

The expected value of a discrete random variable is computed as the ____________________________________.

weighted average of the possible outcomes of that RV where the weights are the probabilities of the outcomes

If X and Y are in fact independent, then cov(X,Y) = ___.

0

If X and Y are uncorrelated, then corr(X,Y) = ___.

0

If a distribution is symmetric, then its skewness = ______.

0

The probabilities of a pdf sum to ______.

1

The joint probability of 2 RVs is the probability that ________________________.

2 RVs simultaneously take on certain values

If a distribution has a long left tail, then its skewness is ___ 0.

<

If a distribution has a long right tail, then its skewness is ___ 0.

>

E(a + bX + cY) = ? Write the formula.

E(a + bX + cY) = a + bE(X) + cE(Y)

True or False. If X and Y are linearly independent, then they are independent.

False

True or False. The kurtosis of a distribution can be positive or negative.

False, it can only be positive

Write the general form of a joint probability distribution.

Pr(X=a , Y=b)

Write the general form of a conditional probability distribution.

Pr(Y=b | X=a)

Pr(Y=b | X=a) = ? Write the formula.

Pr(Y=b | X=a) = [Pr(Y=b , X=a)] / [Pr(X=a)]