StatUoB

Random variables and probability distributions:

• X --> A random variable is a numerical measure of the outcome from a probability experiment, so its value is determined by chance. Typically, denoted as the letter, X.e.g. rolling a dice
• A probability distribution is a table, formula, or graph that describes the values of a random variable and the probability associated with these values.

• Discrete variables produce outcomes that come from a counting process (e.g. number of classes you are taking)
• Continuous variables produce outcomes that come from a measurement (e.g. your annual salary, or your weight).

• In continuous distributions, the probability that X equals to a certain value is zero à Because there is no area.

• The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable.
• Also called a rectangular distribution

density function => 1/(b-a)

bild

• Bell Shaped, Symmetrical, Mean = Median = Mode
• Location is determined by the mean, μ
• Spread is determined by the standard deviation, σ
• The random variable has an infinite theoretical range
• μ ± 1σ encloses about 68% of X’s
• μ ± 2σ covers about 95% of X’s
• μ ± 3σ covers about 99.73% of X’s

Idea: Number of observations that are free to vary after sample mean has been calculated. It increases variation and therefore also standard deviation extremely when n is small and approximates n when n is getting bigger.

Recap: Discrete variables à Variables producing outcomes that come from a counting process.

Rules:

1. A fixed number of observations, n
   1. e.g. 15 tosses of a coin
   2. e.g. 10 light bulbs taken from a warehouse
2. Constant probability for the event of interest occurring (π) for each observation
   1. e.g. Probability of getting a tail is the same each time we toss the coin.
3. Each observation is categorized as to whether the “event of interest” occurred or not.
   1. e.g. head or tail in each toss of a coin
   2. e.g. defective or not defective light bulb
   3. When the probability of the event of interest is represented as π, then the probability of the event of interest not occurring is 1 – π.
4. Observations are independent
   1. The outcome of one observation does not affect the outcome of the other

negatively skewed (LS) when p > 0,5; Symmetric when n = 10 and p = 0,5 and prositively skewed (RS) when p < 0,5

Mean = np

Var = np(1-p)

Sd = sqrt(var)

• A quantitative measure of uncertainty
• A measure of the strength of belief in the occurrence of an uncertain event
• A measure of the degree of chance or likelihood of occurrence of an uncertain event
• Measured by a number between 0 and 1 (or between 0% and 100%)

set: a collection of elements or objects of interest

empty set : a set containing no elements

universal set: a set containing all possible elements

complement (not): the compelement of A is Abar and is a set containing all elements of S not in A.

Intersection AND: a set containing all elements in both A and B AnB

Union( OR): a set containing all elements in A or B. AuB

Mutually exclusive: or disjoint set: Sets having no elements in commen, having no intersection, whose intersection is the empty set.

Partition: a collection of mutually exclusive sets which together include all possible elements, whose union is the universal set. AKA collectively exhaustive.

• Process that leads to one of several outcomes
•  Each trial of an experiment has a single observed outcome
• The precise outcome of a random experiment is unknown before a trial.

Sample Space or Event Set: Set of all possible outcomes (universal set) for a given experiment. --> Roll a six-sided dice S={1,2,3,4,5,6}

Event: Collection of outcomes having a common characteristic. --> Even numbers  A = {2,4,6}.

Probability of an event: Sum of the probabilities of the outcomes of which it consits --> P(A) = P(2) + P84 + P(6)

bild

bild

Order important:

• replace = True = n^k
• replace = false = n!/(n-k)!

Order not important:

• replace =True = (n-1+k)!/(n-1)!k! --> not important
• replace = false = (n k)

Regression analysis is used for 

• explaining the impact of changes in independent variables on the dependent variable.
• predicting the value of a dependent varibale based on the value of independent variable.

Dependent variable --> the variable we wish to predict or explain. --> Y

Independent variable --> the variable used to predict or explain the dependent variable. --> X

The coeffision of determination is the portion of the total veariation in the dependet varibale that is explained by variation in the independent variable

the coefficient of determination is also called r-squared and is denoted as r^2

Linearity: the relationship between x and y is linear 

• Plot the residuals and check if the residual are linear distributed.

Independent:

• Error values are statistically independent
• Durbin-Watson test (dwtest()) p < 5% must be satisfied.

Normality

• Error values are normally distributed for any given value of x
• Examine the Histogram of the residuals
• Construct a Normal Probability Plot of the residuals
  • In Normal Probability Plot, if the QQ plot approximately 
    follows a straight line, the residuals can be regarded as
    normally distributed
• Use Shapiro-Wilk normality test --> p < 5% must be satisfied.

Equal variance (homoscedasticity)

• Use Breusch-Pagan test against heteroskedasticity: e.g. 
  bptest() {lmtest} in R
•  

T

T

Kurtosis affects the peakedness of the curve of the distribution—that is, how sharply the curve rises approaching the center of the distribution

The correlation ρ (= the coefficient of correlation) represents a strength / direction of linear relationship between two variables.

The correlation ρ (= the coefficient of correlation) represents a strength / direction of linear relationship between two variables

The correlation has the following features:

• Unit free
• Range between –1 and 1
• The closer to –1, the stronger the negative linear relationship
• The closer to 1, the stronger the positive linear relationship
• The closer to 0, the weaker the linear relationship

• is most commonly used measure of variation
• shows variation around the mean
• is the square root of the variance
• has the same units as the original data