F2 Mathematical Foundations / Probability Theory I

Question 1

What is the formula for the variance of x

Answer 1

Var(x)=E[(x-my_x )^2]

Question 2

What is the formula for the mean of x

Answer 2

E(x)=(∑x_i )/n

Question 3

What are three types of probability theory?

Answer 3

1) Classical / frequentist
2) Empirical / bayesian
3) Subjective

Question 4

What is the difference between probabilistic and deterministic theories?

Answer 4

Probabilistic: If A then B is likely (used in political science)

Deterministic: If A then B.

Question 5

What is a distribution?

Answer 5

A set of values that a random variable may take. Each values has a weight (=probability)

Question 6

Which of the following a linear operator?
- Variance
- Expected value
- Integral

Answer 6

• Variance (no)
• Expected value (yes)
• Integral (yes)

Question 7

What is a constraint for the CDF?

Answer 7

It is non-decreasing. If x increases then the probability will always increase or be the same.

Question 8

What is a constraint of the PDF?

Answer 8

It is no defined for negativ values
f(x)>0

Question 9

What is an interger?

Answer 9

Integers: {..-2,-1,0,1,2…}

Question 10

What is the difference between a scalar, a vector and a matrix?

Answer 10

Scalar: One value
Vector: One dimensional/combination of values
Matrix: Two dimensional (can also be a vector)

Question 11

What is the difference between an explicit and implicit function?

Answer 11

Explicit: y = x
Implicit: y = f(x)

Question 12

How do you interpret the beta-koefficent?

Answer 12

With Δx what is the average increase in y?

Question 13

What is euler’s number? What is it raised to the power of 1 and 0?

Answer 13

2.7182

e^1 = 2.7182
e^0 = 1

Question 14

What does factorial mean? And what is 0! and 1!

Answer 14

The product of all positive integers from 1 to x (with x!). 5!=54321=120

0! = 1
1! = 1

Question 15

How does the sum symbol work? Draw it

Answer 15

Index of summation: i (integer values)
Lower limit: m
Upper limit: n
The expression to be summed / element: f(i)

How it works:

Initialization: Start with i=m
Iteration: Increment i with 1 after each step
Termination: Stop once i exceeds n
Summation: Add the values of f(i) for each i from m to n

Question 16

What are four relevant words of probability theory?

Answer 16

Element of: ∈
Sample space: Entirety of all outcomes (S)
Outcome: Smallest subunit
Event: Partitions of the sample space (S)

Question 17

What is and intersection and union? Draw a Venn diagram

Answer 17

Intersection (A and B): P(A∩B)
Union (A or B): P(A∪B)

Question 18

How do you write discrete values? How do you indicate whether endpoints are included or not?

Answer 18

Endpoints not included: (0,1)
Endpoints included: [0,1]
Discrete values: {0,1}

Question 19

What is the complement of A?

Answer 19

Everything that is not A

Question 20

How can you calculate the intersection of A and B?

Answer 20

P(A∩B) = P(B|A)P(A) = P(A|B)P(B)

Question 21

How do you calculate the union of A and B?

Answer 21

P(A∪B) = P(A)*P(B) - P(A∩B)

If A and B are independent:
P(A∩B) = P(A)*P(B)

Question 22

Define independence and draw it in a regression

Answer 22

The occurrence of A does not depend on B. Probability of A is independent from B.

P(A|B)=P(A)

Regression line is flat.

Question 23

When are to sets mutually exclusive?

Answer 23

Two events are mutually exclusive when one cannot occur if the other has occurred.

A form of total dependence (no elements in their intersection).

P(A|B)=0
P(A∪B) = P(A) + P(B)

Question 24

What does collective exhaustivity means?

Answer 24

Each and every event fits into at least one of the categories. Not the same as independence!

P(A)+P(B) = 1

Question 25

What is DGP?

Answer 25

Data generating proces

Question 26

What are important laws of probability?

Answer 26

Non-negativity: Probability must be positive
Normalization: The sample space =1

Question 27

What does discrete variabel mean?

Answer 27

It’s countable.

Question 28

What is countable infinity?

Answer 28

There is no upper limit but it’s still counted

Question 29

Define a random variabel.

Answer 29

A variable that can take on array of values with the probability that it can take on any value defined according to a random process (a process we can model)

Question 30

What do we assume about the error term?

Answer 30

Epsilon is normally distributed ε~N(0,σ^2).

Question 31

What is a PMF? What is the input and output?

Answer 31

A probability mass function.

Input: Discrete value
Output: Probability for the value [0,1]

Following the law of normalization all possible outcomes sum to 1

Question 32

What is a CDF for discrete variables? What is the input and output?

Answer 32

A cumulative distribution function - probability summed up to a certain value

Input: Discrete value
Output: Probability of the cumulative probability up until the value [0,1]

Probability will always increase in vertical steps (the staircase).

Question 33

What is a parameter?

Answer 33

A parameter describe the shape of a distribution e.g. the location parameter (mean/my) and the scale/dispersion (variance/sigma).

Question 34

Mention four discrete distributions

Answer 34

Uniform
Poisson
Bernoulli
Binomial

Question 35

What is a uniform distribution?

Answer 35

All events have a equal probability of occurring. Events are mutually exclusive and collectively exhaustive (the dice).

PMF: 1/n

Question 36

Describe the poisson distribution

Answer 36

It’s used for count variables and supported in discrete steps of 1 from 0 to infinity. It has one parameter, my, and my > 0.

The poisson is approximately normally distributed for high values of my.

Question 37

What is a feature of the PMF and the mean

Answer 37

The probability mass is distributed equally on either side of the mean. If the mean is close to zero, then a big amount of the probability mass must be between 0 and 1.

Question 38

Describe the Bernoulli distribution

Answer 38

Single draw of either succes or failure supported {0,1} with one parameter p (probability).

1 = p
0 = 1 - p

Question 39

What is the binomial distribution?

Answer 39

Repeating sessions of the Bernoulli. Two parameters:

n = number of trials (discrete from 0 to infinity)
p = probability of n successes (0,1)

Approximation to the normal distribution is possible.