Agenda Course 02402 Introduction to Statistics 1 Stochastic Variables and Distributions The definition of a Stochastic Variable Lecture 2: Discrete Distributions Density function The distribution function 2 Specific Statistical distributions Per Bruun Brockhoff The Binomial distribution Example 1 The hypergeometric distribution DTU Informatics Example 2 Building 305 - room 110 The Poisson distribution Danish Technical University Example 3 2800 Lyngby – Denmark 3 Mean and Variance e-mail: pbb@imm.dtu.dk Mean values and variances of some known discrete distributions 4 R (R Note section 3 ) Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 1 / 38 Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 2 / 38 Stochastic Variables and Distributions The definition of a Stochastic Variable Stochastic Variables and Distributions The definition of a Stochastic Variable Stochastic Variables From Chapter 3: The classical concept of probability is defined as: A stochastic variable, what is that? If there are n are equally likely possibilities, of which one Stochastic variables are written with capital letters, e.g. must occur and s are regarded as favorable, or as a X, Y, Z ’success’, then the probability of a "success" is given by: The outcome of a stochastic variable is written with the corresponding lower case letters. x, y, z s We differ between discrete and continuous stochastic n variables s=number of favorable outcomes n=number of possible outcomes Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 4 / 38 Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 5 / 38
Stochastic Variables and Distributions Density function Stochastic Variables and Distributions Density function Density function The density function for a discrete variable For a discrete variable the density function can be written The Density function of a stochastic variable is denoted as: by f ( x ) f ( x ) = P ( X = x ) We often call it a frequency function or where: f ( x ) tells something about the frequency of the f ( x ) > 0 for x ∈ S outcome x for the stochastic variable X f ( x ) = 0 for x ∈ S A good plot of f ( x ) is a bar chart (discrete) or a histogram (continuous) � f ( x ) = 1 Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 6 / 38 Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 7 / 38 Stochastic Variables and Distributions The distribution function Specific Statistical distributions The distribution function Specific Statistical distributions The distribution function for a stochastic variable is denoted by F ( x ) . A number of statistical distributions exists that can be The distribution function is equal to the cumulative used to describe and analyse different kind of problems density function: First we consider discrete distributions F ( x ) = P ( X ≤ x ) The binomial distribution The hypergeometric distribution The Poisson distribution x � F ( x ) = f ( t ) t = −∞ Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 8 / 38 Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 10 / 38
Specific Statistical distributions The Binomial distribution Specific Statistical distributions The Binomial distribution The Binomial distribution The binomial distribution We say that a stochastic variable, X , follows a binomial distribution X ∼ b ( x ; n, p ) The density function for the binomial distribution: We consider n independent trials, where n is a constant � n � p x (1 − p ) n − x f ( x ) = P ( X = x ) = In every trial the outcome be either success or failure x The probability of success p (same for all n trials) The distribution function for the binomial distribution: F ( x ) = P ( X ≤ x ) table 1, page 505 (7ed: 576) (6ed: 565) → The distribution function F ( x ) = P ( X ≤ x ) is written as B ( x ; n, p ) Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 11 / 38 Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 12 / 38 Specific Statistical distributions The Binomial distribution Specific Statistical distributions The Binomial distribution The binomial distribution Example 1 In a call center in a phone company the costumer satisfaction is an issue. It is especially important that when errors/faults occur, then they are corrected within the same The binomial distribution is also used to analyse day. samples with replacement Assume that the probability of an error being corrected The hypergeometric distribution is used to analyse samples without replacement within the same is p = 0 . 7 . Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 13 / 38 Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 14 / 38
Specific Statistical distributions The Binomial distribution Specific Statistical distributions The Binomial distribution Example 1 Example 1 During a day, 6 errors are reported. What is the probability What is the probability that at most one of the errors is that all of the errors are corrected within the same day? corrected within the same day? Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 15 / 38 Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 16 / 38 Specific Statistical distributions The Binomial distribution Specific Statistical distributions The hypergeometric distribution Example 2 The hypergeometric distribution In a shipment of 15 hard disks five of them have smal scrathes. We consider: A random sample of 3 hard disks is taken. What is the A population of size N probability that 1 of them has scratches? We take a sample of size n There are a defective population There are N − a none-defective in the population Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 17 / 38 Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 18 / 38
Specific Statistical distributions The hypergeometric distribution Specific Statistical distributions The hypergeometric distribution The hypergeometric distribution The hypergeometric distribution We say that a stochastic variable, X , follows a hypergeometric distribution X ∼ h ( x ; n, a, N ) The density function for the hypergeometric distribution is The hypergeometric distribution can be approximated by the binomial distribution if the population N is big and the � a � � N − a � sample n is small x n − x f ( x ) = P ( X = x ) = � N � n Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 19 / 38 Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 20 / 38 Specific Statistical distributions The hypergeometric distribution Specific Statistical distributions The hypergeometric distribution Example 2 Example 3 In a shipment of 15 hard disks five of them have smal Assume that on average 0.3 patients per day are put in scrathes. hospital in Copenhagen due to air pollution A random sample of 3 hard disks is taken. What is the What is the probability that at most two patients are put in probability that 1 of them has scratches? hospital in Copenhagen due to air pollution on a given day? Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 21 / 38 Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 22 / 38
Specific Statistical distributions The Poisson distribution Specific Statistical distributions The Poisson distribution The Poisson distribution The Poisson distribution The Poisson distribution is often use as distribution X ∼ P ( λ ) (model) for counts which do not have a natural upper The density function: bound f ( x ) = P ( X = x ) = λ x x ! e − λ The Poisson distribution is often characterized as The distribution function: intensity, that is on the form number/unit F ( x ) = P ( X ≤ x ) table 2, page 510 (7ed: 581) (6ed: The parameter λ gives the gives the intensity in the 570) Poisson distribution Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 23 / 38 Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 24 / 38 Specific Statistical distributions The Poisson distribution Specific Statistical distributions The Poisson distribution Example 3 Example 3 Assume that on average 0.3 patients per day are put in What is the probability that exactly two patients are put in hospital in Copenhagen due to air pollution hospital in Copenhagen due to air pollution on a given day? What is the probability that at most two patients are put in hospital in Copenhagen due to air pollution on a given day? Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 25 / 38 Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 2 Fall 2012 26 / 38
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