Statistics Lecture 4 & 5 Random variable •David Bartl •Statistics •INM/BASTA Outline of the lecture •The concept of probability •Random variable •Measures of central tendency (mean, mode, median) •Measures of dispersion (variance) •Measures of shape (skewness, kurtosis) •Functions of random variables (sample mean, sample variance) The concept of probability The concept of probability The concept of probability The concept of probability Random variable (the set of all possible outcomes of a random experiment) (the collection of all measurable events) Random variable Random variable Assumptions to simplify the matters Assumptions to simplify the matters Assumptions to simplify the matters Assumptions to simplify the matters Probability mass function Probability mass function Assumptions to simplify the matters Assumptions to simplify the matters Probability density function Lebesgue measure and Lebesgue integral Lebesgue measure and Lebesgue integral Assumptions to simplify the matters Assumptions to simplify the matters Examples of random variables 0 10 20 30 40 50 60 70 80 90 100 Examples of random variables 100 000 600 000 Examples of random variables time •The “wheel of fortune”. • •The customer rotates the wheel and, depending upon the final position, the discount of the price is deduced. Examples of random variables kolo stesti1 Examples of random variables Examples of random variables •Bar chart – the frequencies (numbers) of the ordinal data item “Discount”: 5 10 0 Discount 20 15 Frequency 25 12 20 30 15 16 25 24 17 12 14 50 3 1 15 1 1 1 70 80 100 Examples of random variables 0 10 20 30 40 50 60 70 80 90 100 0.05 0.10 0 0.20 0.15 0.25 Discount Probability mass function & Probability density function Cumulative distribution function Cumulative distribution function Cumulative distribution function Examples of cumulative distribution functions Examples of cumulative distribution functions The cumulative distribution & the density function The cumulative distribution & the density function The cumulative distribution & the density function The probability of an event 100 000 600 000 250 000 Then the probability the salary of an employee is in the range from 100000 to 250 000, say, is: Measures of central tendency •Mean / Expected value •Mode •Quantile •Median •Quartiles •Deciles •Centiles • The mean / expected value The mode The mode The mode 0 10 20 30 40 50 60 70 80 90 100 0.05 0.10 0 0.20 0.15 0.25 Discount 14 The mode 0 1 2 3 4 5 6 7 8 9 10 The mode 0 1 2 3 4 5 6 7 8 9 10 The mode 0 1 2 3 4 5 6 7 8 9 10 The mode The mode The mode The mode The mode •Remarks: •There may exist more than one mode. •The probability distribution is termed: • — unimodal, if there is exactly one mode • — bimodal, if there are exactly two modes • — etc. • — multimodal, if there are several modes Quantile Quantile Quantile Median Quartiles Quartiles Median & Quartiles lower quartile upper quartile Deciles Centiles Mode & Mean & Median may differ Med(X) E(X) Mod(X)= Measures of dispersion •Variance •Standard deviation • Variance Variance Variance Variance Variance Standard deviation Measures of shape •Skewness •Kurtosis • Skewness & Kurtosis Skewness: Properties and interpretation Kurtosis: Properties and interpretation Functions of random variables •Sample mean •Sample variance •Sample standard deviation • Statistic Sample mean & Sample variance Sample variance & Sample standard deviation The measures of the statistics The expected values of the functions of random variables •The expected value of the sample mean •Independent events •Independent random variables •The variance of the sample mean •The expected value of the sample variance • The expected value of the sample mean Independent events Independent random variables Independent random variables: Theorem Independent random variables: E[XY] = E[X] E[Y] Independent random variables: E[XY] = E[X] E[Y] Independent random variables: Theorem II The variance of the sample mean The variance of the sample mean The variance of the sample mean The expected value of the sample variance The expected value of the sample variance The expected value of the sample variance The expected value of the sample variance The expected value of the sample variance An alternative formula for the sample variance An alternative formula for the sample variance