Summary Introduction data analysis lectures, formulas and book

Week 5

Bernoulli distribution: used to model random variables with two outcomes

. Bernoulli experiment: random experiment with outcomes classified as S (success) and F (failure)

. P(S): probability of success

. Bernoulli random variable:



. P(X=1): often denoted by p: probability of success

. P(X=0) = 1-p: probability of failure

. Notation: X~Bern(p) (X has the bernoulli distribution with parameter p)

. Pdf of X:




. E(X) = p

. V(X) = p(1-p)

. If X~Bern(p) then Y = 1 – X~Bern(1-p)



Binominal and hypergeometric distributions are often used to model a rv X that only has non-negative
integers 0,1...,n as possible outcomes



Binominal experiment with parameter p: random experiment that consists of a series of independent and
identical repetitions (trials) of a Bernoulli experiment

. The outcomes of a binominal experiment with n trials are n-tuples that have S or F on the positions

. Interest is in the rv Y that counts the number of successes

. Y can have the outcomes 0,1,…,n

. Y~Bin(n,p)

. Y has the binominal distribution with parameters n and p

,Calculating probabilities with excel:

. Let Y~Bin(n,p), then:

. f(k) = P(Y=k) = binom.dist(k,n,p,0)

. F(k) = P(Y≤k) = binom.dist(k,n,p,1)



Proportion of successes: consider a binominal experiment with n and p

. Y = #successes

. Proportion of successes:

. Properties:




. If n is very large, V is close to 0



Random experiment: n elements are randomly drawn from a population of size N with M successes and N-
M failures

. Y: number of successes in the sample

. If drawn with replacement:

. Y~Bin(n,p) with p=M/N




. If drawn without replacement: the n trials are not independent so there is no binominal distribution: Y has
a hypergeometric distribution (will not be treated in this course)

. Only has the result:




. If the population size is much larger than the number of draws, it does not mattes much whether
we work with or without replacement



Uniform distributions: used as models for continuous random variables Y with (more or less) equally likely
outcomes in bounded intervals

. Uniform distribution with parameters α and β, α<β

,. Y~U(α,β): Y has the uniform distribution with parameters α and β




Week 6

Family of continuous distribution: normal distribution

Joint probability distribution: discrete joint probability density function



Normal distribution: often used as a model for a random variable that is the result of many independent
contributions

. Plays an important role in hypothesis testing and confidence intervals

. Bijv. height/weight of a randomly selected person, asset returns, investment returns, central Limit
Theorem: Sample mean approximately follows a normal distribution, central Limit Theorem: Sample
proportion approximately follows a normal distribution

, In calculating probabilities, always make a sketch of the graph of the density and indicate the
corresponding area



Calculating probabilities with excel:


P(Y≤y) = norm.dist(y,μ,σ,1)



Calculating p-quantiles with excel:

such that P(Y≤b)=p: use norm.inv(p,μ,σ)



Standard normal distribution: the distribution N(0,1)

. Φ: cdf

. Φ(z) = P(Z≤z)

. For Z~N(0,1) the pdf is symmetric around 0




. For a>0:

. P(Z>a) = P(Z<-a)

. P(-a<Z<a) = 2Φ(a) -1