bg business statistics chapter5 0112

Chapter 5: Discrete probability distributions TRỊNH THỊ HƯỜNG Thuongmai University, Hanoi, Vietnam trinhthihuong@tmu.edu.vn March 12, 2021 T.T.Huong (TMU) Business Statistics March 12, 2021 / 13 Table of contents Statistics in Practice Random variables Discrete Probability Distributions Expected value and variance Binomial probability distribution T.T.Huong (TMU) Business Statistics March 12, 2021 / 13 Statistics in Practice Citibank T.T.Huong (TMU) Business Statistics March 12, 2021 / 13 Random variables Random variables A random variable is a numerical description of the outcome of an experiment A random variable can be classified as being either discrete or continuous depending on the numerical values it assumes Discrete Random Variables A random variable that may assume either a finite number of values or an infinite sequence of values such as 0, 1, 2, A continuous random variable A random variable that may assume any numerical value in an interval or collection of intervals T.T.Huong (TMU) Business Statistics March 12, 2021 / 13 Random variables Figure: EXAMPLES OF DISCRETE RANDOM VARIABLES Figure: EXAMPLES OF CONTINUOUS RANDOM VARIABLES T.T.Huong (TMU) Business Statistics March 12, 2021 / 13 Discrete Probability Distributions Discrete Probability Distributions The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable For a discrete random variable x, the probability distribution is defined by a probability function, denoted by f (x) The probability function provides the probability for each value of the random variable Required conditions for a discrete probability function f (x) ≥ f (x) = T.T.Huong (TMU) Business Statistics March 12, 2021 / 13 Discrete Probability Distributions Example Consider the sales of automobiles at DiCarlo Motors in Saratoga, New York Over the past 300 days of operation, sales data show 54 days with no automobiles sold, 117 days with automobile sold, 72 days with automobiles sold, 42 days with automobiles sold, 12 days with automobiles sold, days with automobiles sold Define the random variable of interest as x is the number of automobiles sold during a day Question: Calculate probability distribution of x? T.T.Huong (TMU) Business Statistics March 12, 2021 / 13 Discrete Probability Distributions Discrete uniform probability function Discrete uniform probability function is f (x) = n where n is the number of values the random variable may have Questions: Give one example about an experiment which follow the uniform probability function T.T.Huong (TMU) Business Statistics March 12, 2021 / 13 Expected value and variance Expected value and variance The expected value, or mean, of a random variable is a measure of the central location for the random variable The formula for the expected value of a discrete random variable x: E (x) = µ = xf (x) Variance of a discrete variables Var (x) = σ = T.T.Huong (TMU) (x − µ)2 f (x) Business Statistics March 12, 2021 / 13 Expected value and variance Excercise Ex 15 to 20, pages 204 - 206 T.T.Huong (TMU) Business Statistics March 12, 2021 10 / 13 Binomial probability distribution A binomial experiment The experiment consists of a sequence of n identical trials Two outcomes are possible on each trial We refer to one outcome as a success and the other outcome as a failure The probability of a success, denoted by p, does not change from trial to trial.Consequently, the probability of a failure, denoted by − p, does not change from trial to trial The trials are independent T.T.Huong (TMU) Business Statistics March 12, 2021 11 / 13 Binomial probability distribution Binomial probability function The probability of x successes in n trials is f (x) = n! p x (1 − p)n−x x!(n − x)! where x the number of success p the probability of a success on one trial T.T.Huong (TMU) Business Statistics March 12, 2021 12 / 13 Binomial probability distribution Excercise Ex 26 and 26 and 29, pages 216 T.T.Huong (TMU) Business Statistics March 12, 2021 13 / 13 ... T.T.Huong (TMU) Business Statistics March 12, 2021 / 13 Random variables Figure: EXAMPLES OF DISCRETE RANDOM VARIABLES Figure: EXAMPLES OF CONTINUOUS RANDOM VARIABLES T.T.Huong (TMU) Business Statistics. .. T.T.Huong (TMU) (x − µ)2 f (x) Business Statistics March 12, 2021 / 13 Expected value and variance Excercise Ex 15 to 20, pages 204 - 206 T.T.Huong (TMU) Business Statistics March 12, 2021 10 /... one trial T.T.Huong (TMU) Business Statistics March 12, 2021 12 / 13 Binomial probability distribution Excercise Ex 26 and 26 and 29, pages 216 T.T.Huong (TMU) Business Statistics March 12, 2021