CHAPTER 2 – DISCRETE DISTRIBUTIONS HÜSEYIN GÜLER MATHEMATICAL STATISTICS Discrete Distributions 1

2.1. DISCRETE PROBABILITY DISTRIBUTIONS The concept of random variable: S: Space or support of an experiment A random variable (r.v.) X is a real valued function defined on the space. X: S → R x: Represents the value of X x ε S X is a discrete r.v. if its possible values are finite, or countably infinite. Discrete Distributions 2

A chip is selected randomly from the bowl: Discrete Distributions 3  S = {1, 2, 3, 4}  X: The number on the selected chip  X is a r.v. with space S  x = 1, 2, 3, 4. X is a discrete r.v. (it takes 4 different values)

P(X = x): Represents the probability that X is equal to x. Discrete Distributions 4 The distribution of probability on the support S The probability mass function (p.m.f.)

Discrete Distributions 5

CALCULATING PROBABILITIES USING P.M.F. Discrete Distributions 6  Compute the probability that the number on the chip is 3 or 4.  If A is a subset of S then

CALCULATING PROBABILITIES USING P.M.F. Discrete Distributions 7  Compute the probability that the number on the chip is less than or equal to 3.

RELATIVE FREQUENCIES AND RELATIVE FREQUENCY HISTOGRAM Discrete Distributions 8  The histogram of relative frequencies is called relative frequency histogram.  Relative frequencies converge to the p.m.f as n increases.  When the experiment is performed n times the relative frequency of x is

The chip experiment is repeated n = 1000 times using a computer simulation. Discrete Distributions 9 x1234Total Frequency n = 1000 h (x)

THE COMPARISON OF f(x) AND h(x) Discrete Distributions 10  f(x) is theoretically obtained while h(x) is obtained from a sample. x1234Total f (x) h (x)

THE MEAN OF THE (PROBABILITY) DISTRIBUTION Discrete Distributions 11 called the mean of X.  It is possible to estimate μ using relative frequencies.  The weighted average of X is

THE MEAN OF THE EMPIRICAL DISTRIBUTION x 1, x 2,..., x n : Observed values of x f j : The frequency of u j u j = 1, 2, 3, 4. Discrete Distributions 12 the empirical distribution the mean of the empirical distribution or the sample mean

THE VARIANCE AND THE STANDARD DEVIATION OF THE DISTRIBUTION Discrete Distributions 13  The variance of X is  The standart deviation of X is

AN ALTERNATIVE FOR THE VARIANCE OF THE DISTRIBUTION Discrete Distributions 14 r_th moment about the origin

THE VARIANCE OF THE EMPIRICAL DISTRIBUTION Discrete Distributions 15

THE VARIANCE AND THE STANDART DEVIATION OF THE SAMPLE Discrete Distributions 16  s 2 (the variance of the sample) is an estimate of (the variance of X).