Topic 1: Stats & Maths Background

Question 1

What is a discrete random variable?

Answer 1

A random variable that can take on a finite, or a countable infinite set of values, with associated propabilities such that:

Question 2

What are the rules for probability distributions?

Answer 2

1. All probabilities lie between 0 and 1.
2. The null set is assigned the probability 0. The full set is assigned 1.
3. The probability assigned toi an event that is the union of two disjoint events is the sum of the probabilities assigned to those disjoint events.

Question 3

What is a CDF?

Answer 3

Cumulative distribution function.

Often denoted F(x), the value at any point is the probabilities that P(X <= x) where X is the random variable.

Question 4

What is a PDF?

Answer 4

A probability density function.

A function such that:

Question 5

How are PDF’s related to CDF’s?

Answer 5

For the PDF, denoted by f(x). (see below).

This is conditional on the CDF being differentiable.

Question 6

What would you expect the CDF and PDF graphs for a normal distribution to look like?

Answer 6

Question 7

How can the following be calculated?

Answer 7

Where F(x) is the CDF of X.

Question 8

What is the first moment of a random variable?

Answer 8

It’s expectation (the mean of the population).

Question 9

How can the expectation of a continous random variable be calculated?

Answer 9

Question 10

What are higher moments of a random variable?

Answer 10

Expectations of the random variable raised to a higher power.

Question 11

What is the formula for calculating the nth moment of a continous random variable X?

Answer 11

Where f(x) is the pdf of X.

Question 12

What is a centered, or central moment? How is it calculated for a CRV?

Answer 12

Rather then taking the value of a variable to some power, the difference between that value and the mean is taken to some power.

Question 13

What is the significance of the second central moment?

Answer 13

It is the variance of the the random variable, denoted

Question 14

Define the CDF of a bivariate distribution

Answer 14

Question 15

How could the CDF of x₁ and x₂ be calculated if they are both statistically independent?

Answer 15

Question 16

How can conditional probabilities be defined?

Answer 16

P

Question 17

What is the formula for conditional density?

Answer 17

Where x₂ != 0

** denominator should be f_X() **

Question 18

What is a conditional expectation?

Answer 18

E(X₁ | x₂)

The expected value of X₁, given X₂ = x₂

Or E(X₁ | X₂),

Which is a function showing the expected value of X₁ given some value of X₂

Question 19

What are some of the properties of conditional expectations, E(X₁ | X₂), where X₂ is a random variable, not a particular value.

Answer 19

1. Law of Iterated Expectations:

E(E(X₁ | X₂ ) ) = E(X)

2.

E(X₂ | X₂) = X₂,

E(X₂² | X₂) = X₂

3.

E(X₁ h(X₂) | X₂) = h(X₂) E(X₁ | X₂) for a deterministic function h

4.

From (3), when E(X₁ | X₂) = 0,

E(X₁h(X₂)) = 0

Question 20

Why is it that E(X₁h(X₂)) = 0 when E(X₁ | X₂) = 0

Answer 20

Question 21

Show the result of the following expression:

Answer 21

Question 22

Show the result of the following Expression:

Answer 22

Question 23

Show by example how matrix multiplication is conducted.

Answer 23

Look at the rows and columns of the first and second matricies respectively, with each element of C, Cij being the matrix multiplication of row i and column j.