GM.Binomia distr - Session 4

BINOMIAL PROBABILITY DISTRIBUTION : 

BINOMIAL PROBABILITY DISTRIBUTION Business Statistics Dr. Gunjan Malhotra Assistant Professor Institute of Management Technology, Ghaziabad gmalhotra@imt.edu mailforgunjan@gmail.com 12/2/2011 1

Probability - Model & Distribution: 

Probability - Model & Distribution A Probability Model is the set of assumptions used to assign probabilities to each outcome in a set of possible outcomes. A Probability Distribution defines the relationship between the outcomes and their probability of occurrence. 12/2/2011 2

Probability Distribution: 

Probability Distribution A probability distribution for a discrete random variable is a mutually exclusive listing of all possible numerical outcomes for that random variable such that a particular probability of occurrence is associated with each outcome. A probability distribution gives the entire range of values that can occur based on an experiment. 12/2/2011 3

Characteristics of a probability distribution : 

Characteristics of a probability distribution The outcomes are mutually exclusive and collectively exhaustive events. Probability of one event is between 0 and 1, inclusive 3. Sum of probabilities of all events is 1. 12/2/2011 4

Types of Probability Distribution: 

Types of Probability Distribution There are two fundamental types of probability distributions Discrete Continuous 12/2/2011 5

Discrete Probability Distributions : 

Discrete Probability Distributions These are used to model a situation where The number of events (things that can happen) is countable (i.e., can be placed in correspondence with integers) Probabilities sum to 1 Each event has a probability 12/2/2011 6

Continuous Probability Distributions : 

Continuous Probability Distributions These are used to model situations where the number of things that can happen is not countable. Probability of a particular outcome cannot be defined. Interval probabilities can be defined. 12/2/2011 7

Discrete Random Variable : 

Discrete Random Variable Expected Value (or mean) of a discrete distribution (Weighted Average) Example: Toss 2 coins, X = # of heads, compute expected value of X: E(X) = (0 x 0.25) + (1 x 0.50) + (2 x 0.25) = 1.0 X P(X) 0 0.25 1 0.50 2 0.25 12/2/2011 8

Discrete Random Variable : 

Variance of a discrete random variable Standard Deviation of a discrete random variable where: E(X) = Expected value of the discrete random variable X X i = the i th outcome of X P(X i ) = Probability of the i th occurrence of X Discrete Random Variable 12/2/2011 9

Types of Theoretical Distribution: 

Types of Theoretical Distribution Binominal distribution Multinomial distribution Negative binomial distribution Poisson distribution Hyper geometric distribution Normal distribution Exponential Distribution 12/2/2011 10

Binomial / Bernoulli distribution: 

Binomial / Bernoulli distribution James Bernoulli [1654-1705] 2 features Where probability is prominent When one event is repeated under the same conditions 12/2/2011 11

Binomial Distribution: 

Binomial Distribution Four Properties of a Binomial Experiment 3. The probability of a success, denoted by p , does not change from trial to trial. 4. The trials are independent. 2. Two outcomes, success and failure , are possible on each trial. 1. The experiment consists of a sequence of n identical trials. 12/2/2011 12

Binomial Distribution: 

Binomial Distribution Our interest is in the number of successes occurring in the n trials. We let x denote the number of successes occurring in the n trials. 12/2/2011 13

Binomial Distribution: 

Binomial Distribution Probability function Mean value Variance and standard deviation 12/2/2011 14

Binomial Distribution Formula: 

P(X) = probability of X successes in n trials, with probability of success p on each trial X = number of ‘successes’ in sample, (X = 0, 1, 2, ..., n ) n = sample size (number of trials or observations) p = probability of “success” P(X) n X ! n X p (1- p ) X n X ! ( ) ! = - - Example: Flip a coin four times, let x = # heads: n = 4 p = 0.5 1 - p = (1 - 0.5) = 0.5 X = 0, 1, 2, 3, 4 Binomial Distribution Formula 12/2/2011 15

Calculating a Binomial Probability: 

Calculating a Binomial Probability What is the probability of one success in five observations if the probability of success is .1? X = 1, n = 5, and p = 0.1 12/2/2011 16

Binomial Distribution: 

Binomial Distribution E ( x ) =  = 3(.1) = .3 employees out of 3 Var( x ) =  2 = 3(.1)(.9) = .27 Expected Value Variance Standard Deviation 12/2/2011 17

Practice Questions: 

Practice Questions Lind – Pg 188 – Q 4 . Pg – 196 – Q 10, 14 , 16 , 18 Pg – 215 – Q 48 , 50, Pg – 217 – Q 62 , 12/2/2011 18