Distributions

Question 1

How are Y and y defined?

Answer 1

Y –> Actual outcome of an event
y –> One of the possible outcomes

Ways of writing the likelihood of a particular outcome y:

P (Y = y)
p (y)

Question 2

What is p(y) called?

Answer 2

Since p(y) expresses the probability of each distinct outcome, we call this:

The probability function

Question 3

With what do we define distributions?

Answer 3

2 characteristics:

Mean –> Average value –> μ

Variance –> How spread out the data is –> σ^2

Question 4

How are population and sample data defined?

Answer 4

Population data –> All the data

Sample data –> Just a part of it

Question 5

How are sample mean and variance denoted?

Answer 5

Sample mean symbol: x̄

Sample variance: s^2 (square)

Question 6

How is the standard deviation defined/denoted

Answer 6

Standard deviation –> Square root of variance:

√(σ^2)

Question 7

Formule standaarddeviatie

Answer 7

Standaarddeviatie:

Sx = σ = de standaarddeviatie van getallenreeks x.
Xi = de waarde van getal i in de getallenreeks.
μ = het gemiddelde van de getallenreeks (som getallen / aantal)
Nx = het aantal getallen in de proef.

Formule Standaarddeviatie
σ = Sx = √( ∑ ( (xi - μ)2 / nx) )

Question 8

Notation for distributions:

Answer 8

Variable name
Tilde sign
Type –> Capital letter
Characteristics (μ, σ^2)

X ~ N (μ, σ^2)

Question 9

Discrete distribution when all outcomes are equally likely?

Answer 9

Equiprobable –> Uniform distribution

Question 10

Discrete distribution with only two possible outcomes?

Answer 10

Follow a Bernoulli distribution
Single trial

Question 11

Discrete distribution when carrying out a similar experiment several times in a row

Answer 11

Binomial Distribution
Two outcomes per iteration
Many iterations –> Multiple trials

Question 12

Discrete distribution when calculating chance of succes after given an average probability?

Answer 12

Poisson distribution

How unusual is an event frequency for a given interval

Example: There’s 35 points per game. How big is the chance of 12 points in the first quarter of the next game?

Question 13

Characteristics of normal distribution?

Answer 13

Often observed in nature
Margin values are called outliers

Question 14

When is Student’s T distribition used?

Answer 14

A small sample approximation of a Normal distribution

It accommodates extreme values much better

Curve has fatter tails than normal distribution

Question 15

When is Chi-Squared distribition used?

Answer 15

A-symmetric continuous distribution

Only consists of non-negative values

Starts from 0

Often used in hypothesis testing

Question 16

When is exponential distribution used?

Answer 16

Present with events that are rapidly changing early on

Example is how online news articles get hits –> Typically when topic is still fresh and then it dies off.

Question 17

When is logistic distribution used?

Answer 17

Logistic distribution

Useful in forecast analysis

Useful for determining a cut-off point for a succesful outcome

Question 18

Discrete distributions characteristics

Answer 18

FINETELY many distinct outcomes

Every unique outcome has a probability assigned to it

Example: Darts board –> The possible outcomes are from 0 to 60, thus finite

Question 19

How do continuous distributions differ from discrete?

Answer 19

Infinitely many consecutive outcomes

Therefore cannot record frequency of each distinct value

Cannot respresent them with a table but with a graph

f(y) >= 0

Question 20

Characteristics of the PDF Graph?

Answer 20

The function is called PDF (Probability Density Function)

Curve is called “Probability distribution curce (PDC)

It is like a discrete distribution but with infinite amount of samples

It gives the probability on y-axis for every possible value ‘y’ on x-axis

Question 21

Likelihood of each individual outcome in continuous distribution?

Answer 21

Likelihood of each individual one is infinitely small

favourable / sample space = 1 / infinite amount

Thus the probability for any individual value is equal to 0 –> P(X) = 0

Also: P(x > X) = P(x >=X)

Example of this: Finishing a run in exactly 6 minutes is extremely unlikely therefore we say the time until 6 minutes is the same as the time including the six minute exact moment.

Question 22

What is the CDF? How is it build up?

Answer 22

When you integrate the PDF, you would get the CDF

When you derive the CDF you would get the PDF

Therefore the CDF runs from 0 to 1

Question 23

What are the characteristics of the normal distribution?

Answer 23

Bell shaped

μ is the mean

Symmetric

Expected value E(X) = μ

Question 24

How would you transform a distribution?

Answer 24

Plus/Minus –> Moves the graphs to the right/left

Multiply/Divide –> It will shrink/expand

Question 25

What is standardizing?

Answer 25

Special kind of transformation where E(x) = 0

Var(X) = 1

The distribution we get after standardizing any normal distribition is called A ‘Standard normal distribution’

Question 26

What is the standard normal distribution table?

Answer 26

There’s a CDF table with all the standardized values for this graph

Also called Z- Score table

Question 27

What is the process of standardizing using transformation?

Answer 27

Moving the vertical centerline to y-axis by adding or subtracting a constant value –> -μ

We need to make sure the standard deviation is 1 –> Divide every element by the value of the standard deviation –> / σ

Question 28

How to calculate the Z-score (based on transformation)

Answer 28

Z = (Y - μ) / σ

Question 29

Define student’s T distribution

Answer 29

t (k) –> Where k is degrees of freedom

Y ~ t(3) –> Variable Y follows t distribution with 3 degrees of freedom

Question 30

When do you use student’s T distribution and what are key difference with normal disribution

Answer 30

You use this when there’s not sufficient data for the normal distribution

Small sample size approximation of a Normal Distribution

Another key difference is that apart from mean and variance you must also define degrees of freedom for the distribution

Graph is also bell shaped but with larger tails to accomodate occurence of values for away from the mean

Question 31

Difference statistics vs characteristics

Answer 31

Statistics:
Sample
60% of 1000 people have brown eyes –> Statistic

Characteristics:
Population
If 65% of people worldwide have brown eyes then that is a characteristic of the human population

Question 32

How can you determine the distribution type and what can you do with it?

Answer 32

Shape of the curve
Characteristics mu and sigma

You can create models (like regression models)

Question 33

How do statistics relate to data science

Answer 33

An expansion of probability, statistics and programming that implements computational technology to solve more advanced questions

Question 34

How does ML fits in the statistics world?

Answer 34

ML relies on expected values a lot.

ML is trial and error where a computer adjusts its expected value along the way

There is always a probability of failure due to unforeseen events (earthquakes etc)