distributions

Inference

how can we astimate the parameters of the population based on the sample

Random Variable properties

Expectation E(X) theoretical mean of entire population

Variance Var/X) theoretical variance of the entire population

Density distribution → for continous RV

probability distribution → for discrete RV

Cumulative distribution function CDF for discrete RV

CDF for continous RV

Binomial verteilung

discrete RV

number of sucesses in independent trails (L), each success has probability p

exp: DNA polymerase

poisson distribution

discrete RV

number of events in time period for a given rate λ of events

exp: number of drops per tile during a rainshower

rate needs to be constant and independent of previous period

negative binomial Verteilung

discrete RV

independent trails, constant probability p

possible questions:

number of successes in L trails → binomial distribution

waiting time until r successes reached → negative binomial distribution with (r,p)

Variance of negarive binomial distribution

can be made arbitrarily wide through making p smaller

variance larger than mean

cannot be described by poisson or binomial process

normal distribution

continous RV

expectation μ, standard deviation σ

Standard normal distribution N(0,1)

μ = 0, σ = 1

transformation of nomal distribution into SND

z-transformation

Properties of SND

\[-1.96, 1.96\] = 95%

\[-1.64, 1.64\] = 90%

critical values

t95 = 1.96

t90 = 1.64

t-distribution

continous RV

distribution of mean values over small samples

Parameters: degree of freedom (df)

→ for lager df towards SND