2- Statistics & OLS with matrices

Question 1

How would you define a random variable?

Answer 1

A numerical summary of random outcomes

Question 2

What is a cumulative probability distribution function?

Answer 2

Gives the probability that a discrete random variable is less than or equal to a certain outcome

Question 3

What is a probability density function?

Answer 3

Probability distribution of a random variable

Question 4

What are the 3 conditions a probability density function must satisfy?

Answer 4

-Nonnegative
-Integrate to 1 (sum of probs is 100%)
-Integral between 2 values equals probability of x being between the 2 values

Question 5

What are moments?

Answer 5

Moments describe certain characteristics of a random variable and its probability function e.g. mean, variance

Question 6

What is mean?

Answer 6

Central tendency of a variable’s density function

Question 7

What is variance?

Answer 7

Measure of spread of a probability distribution

Question 8

How can mean be expressed mathematically?

Answer 8

E(X) or integral of xf(x) between both infinities

Question 9

2 ways variance can be expressed in terms of expected value?

Answer 9

E(X²) - E(X)²
E(xx’) - E(x)E(x)’

Question 10

What is the beta OLS estimator in matrix form?

Answer 10

(X’X)⁻¹X’y

Question 11

What is the OLS estimator in vector form?

Answer 11

x’y/x’x

Question 12

What is var(x + y)?

Answer 12

var(x) + var(y) + 2cov(x,y)

Question 13

What is cov(x,y) in terms of expected values?

Answer 13

E(xy’) - E(x)E(y)’

Question 14

How do you know an expression is Fixed i.e. no longer random?

Answer 14

Expected values are fixed

Question 15

If x has mean μ E(x)=μ, what is the mean of x/v E(x/v)?

Answer 15

μ/v

Question 16

What is a key property for the OLS estimator to be unbiased?

Answer 16

E(u) = 0