Probability Distributions

Question 1

What is the purpose of Probability?

Answer 1

Probability is a measure that quantifies randomness and allows us to draw inferences about quantities and hypotheses of interest.

Question 2

Probability Axioms (Kolmogorov)

Answer 2

1. Probabilities cannot be negative.
2. The probabilty of the sample space is equal to 1. (we know with certainty that outcomes occur, but we do not know for sure which ones)
3. The peobabilty of the union if mutually exclusive events is ewual to the sum of the probabilities of the individual events.

Question 3

What is a random variable?

Answer 3

We obtain a random variable when we assign distinctive nunerical values to the sample points. A random variable is a variable whose values are subject to chance and therefore not known a priori.

Question 4

What is a constant?

Answer 4

Constants only have one distinctive value and it will be realized with the probability of 1.

Question 5

What is a fixed variable?

Answer 5

There are multiple distinctive values but their realization is not subject to chance (e.g. the researcher has full control and hands out values)

Question 6

What types of random variables are there?

Answer 6

Discrete Variables and Continuous Variables

Question 7

What is a discrete Variable?

Answer 7

The variable takes on only discrete values; between two adjacent values, no other outcomes are defined.

Bsp. Number of protests in a particular year

Question 8

What is a continuous variable?

Answer 8

The variable can take on any value; between twi adjacent values, an infinitely large numver of outcomes is defined.

Bsp. The duration of coalition governments

Question 9

What is a Probability Distribution?

Answer 9

A real-valued function describing the probability of a randol variable taking on a certain discrete value or range of values.

-> list of outcomes and their associated probabilities.

Question 10

What types of Probability Distributions are there?

Answer 10

1. Probability Mass Function for discrete RVs.

2. Probability Density Function for continuous RVs.

Question 11

What Attributes do Probability Distributions have?

Answer 11

1. The support of the distribution (values it can take on)
2. The parameters of the distribution:
   - Location
   - Scale
   - Shape

Question 12

What is the support of a distribution?

Answer 12

The range of values of Y for which f(y) is non-zero.

Question 13

What is a location parameter?

Answer 13

It shifts the distribution over its support (rechts - links); e.g. the mean

Question 14

What is a scale parameter?

Answer 14

It influences the spread of the distribution: smaller variance -> more narrow; wider variance -> more wide; e.g. range

Question 15

What are shape parameters?

Answer 15

They influence other aspects of the distribution (e.g. sudden drop or stetig)

Question 16

How can the Probability Mass Function be described?

Answer 16

f(y) = Pr(Y = y) -> Outcome: Probability

Y = random variable
y = realization of variable

The function complies with the following axioms from probability theory:

1. f(y) cannot be negative
2. The probability of the sum of the sample spaces of the function is equal to 1.

Bsp. Würfel: f(x) = 1/6

Question 17

How can the Probility Density Function be described?

Answer 17

Integral von f(y)dy = Pr(a<=y<=b)
Outcome: Area under a curve that represents a probability

The function complies with the following axioms:

1. f(y) cannot be 0
2. Sample space is equal to 1.

probability of one certain event is 0 because summing up all infinite pieces must be 1.

Question 18

What is the Cumulative Distribution Function?

Answer 18

F(y) = Pr(Y<=y)

= Probability of scorinh a value y or a smaller value

The CDF is

• bounded between 0 and 1
• a non-decreasing function in y

How probable is it that we obtain a max value?
discrete RV: Sum of f(y) (adding up)
continuous RV: Integral

Question 19

What is the relationship between CDF and PDF?

Answer 19

The PDF is the first derivative of the CDF; the CDF is the antiderivative of the PDF

Question 20

What are the three main functions of probability distributions?

Answer 20

1. statistical: statistical inference, description of the sampling distributions of statistics
2. empirical: provide mathematical descriptions for empirical patterns
3. theoretical: provide mathematical descriptions of how we beliebe a phenomenon to operate (assumptions), allows statistical, parametric inference

Question 21

What is a Bernoulli distribution?

Answer 21

1 is success, 0 is failure
one single parameter is the probability of success

Bspy flip a fair coin

Question 22

What is a Binomial Distribution?

Answer 22

builds on Bernoulli distribution, but has two parameters n and probability of sucess

Bsp. roll a fair die several times

Question 23

What is a Normal Distribution?

Answer 23

It has two parameters M (mean) and o (sd) > 0

standard normal distribution: M=0, o=1

Question 24

what is the student‘s t distribution?

Answer 24

only one parameter (degrees of freedom), as n increases it becomes indistinguishable from the normal distribution

Question 25

What is the poisson distribution?

Answer 25

one parameter, variance and mean are the same; used to model event counts

Question 26

What is the weibull distribution?

Answer 26

two parameters, event duration models

Question 27

What is the difference between probability and likelihood?

Answer 27

If we have a distribution with specific known parameters and are interested in knowing what data outcomes we can expect given these parameters, we are looking for the probability of these outcomes. Probability is consitioned on parameters and is interested in the outcomes that these are mist probably going to produce.
If we have a specific data set and are interested in knowing what parameters we can expect to best fit the distribution of this data, we are looking for the likelihood od these parameters. Likelihood is conditioned on a set of outcomes and is interested in the parameters that fit it best.