1. Random Variables and Probability Theory

Question 1

In empirical analysis why is there randomness?

Answer 1

Because empirical models can’t capture all relationships

Question 2

Random variable

Answer 2

Any variable whose value is a real number that can’t be predicted exactly , and can be viewed as the outcome of chance

Question 3

PDF

Answer 3

A function that governs how probabilities are assigned to interval values for a random variable

Question 4

CDF

Answer 4

CDF for a random variable x gives the probability that c will take a value less than equal to a specified value. It is a monotonically increasing function of the PDF

Question 5

How are the PDF and CDF related?

Answer 5

The PDF is a derivative of CDF

Question 6

Joint PDF

Answer 6

Gives the probability of two random variables will fall in a specified interval

Question 7

What do we call it when we get the PDF of x or y from the joint distribution?

Answer 7

The marginal PDF

Question 8

Conditional PDF

Answer 8

When we are interested in the distribution of one random variable given the other variable takes a certain value

Question 9

When will the conditional distributions of two random variables be the same as the corresponding marginal distributions

Answer 9

When the random variables are independent

Question 10

Statistical independence

Answer 10

One event occurring has no effect on the prob of the other event occurring

Question 11

How can we check for independence of two variables

Answer 11

If the joint PDF = the product of the marginal PDFs
f(x,y) =f(x)f(y)

Question 12

Mean

Answer 12

A random variables average value

Question 13

Variance

Answer 13

Measures the random variables dispersion around the average value

Question 14

What is the E(x) for a uniformly distributed random interval?

Answer 14

The midpoint of the interval

Question 15

What is the expected value of a sum of random variables equal to?

Answer 15

The sum of their expected values
E(X+Y) = E(X) + E(Y)

Question 16

How can variance be worded?

Answer 16

The average squared deviation of x from its mean.
Or
The expected value of x^2 minus the squared expected value of x

Question 17

What happens to the variance if all values of x are multiplied by a constant a?

Answer 17

The variance is multiplied by a^2

Question 18

Covariance

Answer 18

Defined as the expected value of the product of their deviation from their individual expected values

Question 19

How can the covariance be written in terms of expected values?

Answer 19

C(X,Y) = E(XY) -E(X)E(Y)

Question 20

Correlation

Answer 20

A standardised measure of covariance which provides a unit free measure of the strength of linear association between x and y

Question 21

How is correlation calculated?

Answer 21

The covariance divided by standard deviation of each variable

Question 22

Properties of covariance

Answer 22

1. Variance of a sum of random variables equals the sum of variances plus two times the covariance V(X+Y) = V(X) +V(Y) +2C(X,Y)
2. If X and Y are independent random variables covariance equals zero (by definition of independence)

Question 23

How can the variance be calculated from the expected value?

Answer 23

Var(Y) = E(Y^2)- (E(Y))^2