Review Chapter - Part 1: Population Statistics

Front 1

What are the two kinds of random variable?

Back 1

Discrete and continuous.

Front 2

What is meant by the expected value of a random variable?

Back 2

The weighted average of the potential outcomes, weighted by their probabilities.

Front 3

Give another name for the expected value of a random variable.

Back 3

The population mean.

Front 4

How do you go about calculating the expected value of a function of a discrete random variable, X?

Back 4

You apply the transformation to every possible value of X and take the weighted average of those.

Front 5

Give the first expected value rule.

Back 5

E(X + Y + Z) = E(X) + E(Y) + E(Z)

Front 6

Give the second expected value rule.

Back 6

E(bX) = b E(X)

where b is a constant

Front 7

Give the third expected value rule.

Back 7

E(b) = b

where b is a constant

Front 8

Prove expected value rule 2.

Back 8

E(bX) = Σ b Xi pi

= b Σ Xi pi

= b E(X)

Front 9

Give the formula for the population mean of a discrete random variable.

Back 9

μx = Σ xi pi

Front 10

Give the formula for the population variance of a discrete random variable.

Back 10

σx^2 = Σ (xi - μ)^2 pi

Front 11

Give another, often easier to calculate, formula for the population variance of a discrete random variable.

Back 11

σx^2 = E(X^2) - μx^2

Front 12

Prove σX^2 = E(X^2) - μ^2

Back 12

σX^2 = E {(X - μ)^2}

= E(X^2 - 2Xμ + μ^2)

= E(X^2) - E(2Xμ) + E(μ^2)

= E(X^2) - 2μ E(X) + μ^2

= E(X^2) - 2μ^2 + μ^2

= E(X^2) - μ^2

Front 13

What is the expected value of the disturbance term?

Back 13

Zero.

Front 14

Prove that the expected value of the disturbance term is 0.

Back 14

X = μ + u

∴ u = X - μ

E(u) = E(X - μ)

= E(X) + E(-μ)

= μ - μ

= 0

Front 15

The population variance of the disturbance term is equal to...

Back 15

...the population variance of the random variable.

Front 16

Prove that σX^2 = σu^2

Back 16

σX^2 = E{(X - μX)^2}

= E(u^2)

since u = X - μX

σu^2 = E{(u - μu)^2}

=E(u^2)

since μu = 0

Front 17

What is a probability density function?

Back 17

A probability density function, is a function that depicts the probability density of a continuous random variable (if it can be written as a function).

Front 18

Give the formula for the population covariance of two random variables.

Back 18

cov(X,Y) = σXY = E{(X - μX)(Y - μY)}

Front 19

Give the formula for the expected value of a continuous random variable.

Back 19

E(X) = ∫ X f(X) dx

where f(X) is the probability density function

and the integration is performed over the range for which X is defined.

Front 20

Give the formula for the expected value of a function of a continuous random variable.

Back 20

E[g(X] = ∫ g(X) f(X) dx

Front 21

Give the formula for the population variance of a continuous random variable.

Back 21

σX^2 = E{(X - μx)^2}

= ∫ (X - μx)^2 f(X) dx

Front 22

What is the difference between a measure of association and a measure of correlation?

Back 22

A measure of correlation controls for the unit of measurement so as to determine a numerically-significant level of relatedness (e.g. normalised between -1 and 1).

Front 23

How do you establish the independence of two random variables?

Back 23

If E[g(X)h(Y)] = E[g(X)] E[h(Y)]

then they are independent since

E(X - μX)E(Y - μY) = [E(X) - μX][E(Y) - μY] = 0 x 0

Front 24

Give covariance rule 1.

Back 24

If Y = V + W,

cov(X, Y) = cov(X, V) + cov(X, W)

Front 25

Give covariance rule 2.

Back 25

If Y = bZ, where b is a constant and Z is a variable,

cov(X, Y) = b cov(X, Z)

Front 26

Give covariance rule 3.

Back 26

If Y = b, where b is a constant,

cov(X, Y) = 0

Front 27

Prove covariance rule 1.

Back 27

If Y = V + W, μY = μV + μW

cov(X, Y) = E{(X - μX)(Y - μY)}

=E{( X - μX )( [V + W] - [μV + μW] )}

Front 28

Prove covariance rule 2.

Back 28

If Y = bZ, μY = bμZ

cov(X, Y) = E{( X - μX )( Y - μY )}

= E{( X - μX )( bZ - bμZ )}

= b E{( X - μX ) (Z - μZ )}

= b cov(X, Z)

Front 29

Prove covariance rule 3.

Back 29

If Y = b, μb = b

cov(X, Y) = E {(X - μX)(Y - μY)}

= E{( X - μX )( b - b )}

= E{0}

= 0

Front 30

Give variance rule 1.

Back 30

If Y = V + W,

var(Y) = var(V) + var(W) + 2 cov(V, W)

Front 31

Give variance rule 2.

Back 31

If Y = bZ,

var(Y) = b^2 Var(Z)

Front 32

Give variance rule 3.

Back 32

If Y = b, where b is a constant,

var(Y) = 0

Front 33

Give variance rule 4.

Back 33

If Y = V + b, where b is a constant,

var(Y) = var(V)

Front 34

Prove variance rule 1.

Back 34

if Y = V + W,

var(Y) = cov(Y, Y) = cov(Y, [V+W])

= cov(Y, V) + cov(Y, W)

= cov([V+W], V) + cov([V+W], W)

= cov(V, V) + 2cov(V, W) + cov(W, W)

= var(V) + var(W) + 2cov(V, W)

Front 35

Prove variance rule 2.

Back 35

if Y = bZ, where b is a constant,

var(Y) = cov(Y, Y) = cov(bZ, Y) = b cov(Z, Y)

= b cov(Z, bZ) = b^2 cov(Z, Z) = b^2 var(Z)

Front 36

Prove variance rule 3.

Back 36

if Y = b, where b is a constant,

var(Y) = cov(b, b) = 0

since b = μb

∴ b - μb = 0

Front 37

Prove variance rule 4.

Back 37

if Y = V + b, where b is a constant

var(Y) = var (V + b)

= var(V) + var(b) + 2 cov(V, b)

= var(V) + 0 + 0

Front 38

Give the formula for the population correlation coefficient.

Back 38

ρXY = σXY / √(σX^2 σY^2)