Lesson 6 R Discrete Probability Distributions Review Objectives

Lesson 6 - R Discrete Probability Distributions Review

Objectives • Understand when a probability experiment follows one of the processes for discrete random variables: – – – Uniform (can be continuous as well) Binomial Geometric Poisson Hypergeometric Negative Binomial • Compute probabilities of discrete random variables • Find the mean and standard deviation of a discrete random variables

Vocabulary • None new

English Phrases Math Symbol ≥ At least > More than < Fewer than ≤ No more than = Exactly ≠ Different from English Phrases No less than Greater than Less than At most Equals Greater than or equal to Less than or equal to Is ∑P(x) = 1 P(X) Cumulative probability or cdf P(x ≤ A) Values of Discrete Variable, X cdf(x > A) = 1 – P(x ≤ A) X=A

Discrete PDF Rules Let P(x) denote the probability that the random variable X equals x, then 1) The sum of all probabilities of all outcomes must equal 1 ∑ P(x) = 1 2) The probability of any value x, P(x), must between 0 and 1 0≤ P(x) ≤ 1

Mean & Variance Mean value of a Discrete Random Variable: Given by the formula μx = ∑ [x ∙P(x)] (Expected Value = μx) where x is the value of the random variable and P(x) is the probability of observing x Variance and Standard Deviation of a Discrete Random Variable: The variance of a discrete random variable is given by: σ2 x = ∑ [(x – μx)2 ∙ P(x)] = ∑[x 2 ∙ P(x)] – μ 2 x and standard deviation is √σ2

Discrete PDFs Uniform: probability is equal distributed Binomial: number of successes in n trials Geometric: number of trials until a success or failure is observed Hypergeometric: sampling without replacement in a small population Negative Binomial: number of trials necessary to observe r# of successes (Geometric is a special case with r=1) Poisson: number of successes in a interval of fixed length (time, distance, or area)

Binomial Experiment Criteria 1) Experiment is performed a fixed number of times. Each repetition is called a trial. 2) The trials are independent 3) For each trial there are two mutually exclusive (disjoint) outcomes: success or failure 4) The probability of success is the same for each trial of the experiment

Binomial Notation The probability of obtaining x successes in n independent trials of a binomial experiment, where the probability of success is p, is given by: P(x) = n. Cx px (1 – p)n-x, x = 0, 1, 2, 3, …, n Where n is # of independent trials of the experiment p denotes the probability of success 1 – p is the probability of failure x denotes the number of successes in n independent trials of the experiment. So 0 ≤ x ≤ n Determining probabilities using calculator: 2 nd VARS: binompdf(n, p, x) or binomcdf(n, p, x)

Binomial PDF and Law of Large Numbers As the number of trials n in a binomial experiment increases, the probability distribution of the random variable X becomes bell shaped. As a rule of thumb, if np(1 -p) ≥ 10, the probability distribution will be approximately bell shaped. (which means the Empirical Rule can be used!!)

Geometric Criteria An experiment is said to be a geometric experiment provided: 1. Each repetition is called a trial. 2. The trials are independent 3. For each trial there are two mutually exclusive (disjoint) outcomes: success or failure 4. The probability of success is the same for each trial of the experiment

Geometric Notation The geometric distribution addresses the number of trials necessary before the first success. If the trials are repeated k times until the first success, we will have had k – 1 failures. If p is the probability for a success and q (1 – p) the probability for a failure, the probability for the first success to occur at the kth trial will be (where x = k) P(x) = p(1 – p)x-1, x = 1, 2, 3, … Using the calculator: 2 nd VARS geometpdf(p, k) for P(X=k) Using the calculator: 2 nd VARS geometcdf(p, k) for P(k ≤ X)

Hyper-Geometric Criteria An experiment is said to be a hyper-geometric experiment provided: 1. The experiment is performed a fixed number of times. 2. Each repetition is called a trial. 3. The trials are dependent 4. For each trial there are two mutually exclusive (disjoint) outcomes: success or failure Note: One of the conditions of a binomial distribution was the independence of the trials so the probability of a success is the same for every trial. If successive trials are done without replacement and the sample size or population is small, the probability for each observation will vary.

Hyper-Geometric Notation The formula for the hyper-geometric distribution is: Np. Cx N(1 -p)Cn-x P(x) = -------------NC n x = 0, 1, 2, 3, …, n Where N is the size of the population, p is the proportion of the population with a certain attribute (success), x is the number of individuals from the population selected in the sample with the attribute and n is the number selected to be in the sample (n-x is the number selected who do not have the attribute)

Negative Binomial Criteria An experiment is said to be a negative binomial experiment provided: 1. Each repetition is called a trial. 2. For each trial there are two mutually exclusive (disjoint) outcomes: success or failure 3. The probability of success is the same for each trial of the experiment 4. The trials are independent 5. The trials are repeated until r successes are observed, where r is specified in advance.

Negative Binomial Notation When we studied the Binomial distribution, we were only interested in the probability for a success or a failure to happen. The negative binomial distribution addresses the number of trials necessary before the rth success. If the trials are repeated x times until the rth success, we will have had x – r failures. If p is the probability for a success and (1 – p) the probability for a failure, the probability for the rth success to occur at the xth trial will be P(x) = … x-1 Cr-1 pr(1 – p)x-r x = r, r+1, r+2, Where r number of successes is observed in x number of trials of a binomial experiment with success rate of p

Poisson Criteria Poisson distributions can be used to computer probabilities of experiments in which the random variable X counts the number of occurrences (successes) of a particular event within a specified interval (usually time or space). An experiment is said to be a Poisson process provided: 1) Probability of two or more successes in any sufficiently small sub-interval* is 0. 2) The probability of success is the same for any two intervals of equal length 3) The number of successes in any interval is independent of the number of successes in any other interval provided the intervals are not overlapping

Poisson Notation If X is the number of successes in an interval of fixed length t, then probability formula for X is (λt)x -λt P(x) = ----- e x! x = 0, 1, 2, 3, … where λ (the Greek letter lamda) represents the average number of occurrences of the event in some interval of length 1 and e = 2. 71828. . (Euler's constant) Using the calculator: Poissonpdf(μ, x) & Poissoncdf(μ, x)

Discrete Probability Functions Summary Name Binomial PDF P(x) = n. Cxpx(1 -p)n-x x= x = 0, 1, 2, … μ σ np √np(1 -p) 1/p √ 1 -p Example: Coin flips (x = number of heads in n=10 trials) Geometric P(x) = p(1 – p)x-1 x = 1, 2, … Example: number of trials necessary before the first success How many flips before the first tail is observed? Np. Cx N(1 -p)Cn-x Hyper. P(x) = ----------x = 0, 1, 2, … np geometric C N n √ (np(1 -p)(N-n)/N-1) Example: Small number sampling without replacement What is P(x=6) women will be selected from population of 20 men & 20 women (N=40) in a n=10 random sampling? Negative Binomial P(x) = x-1 Cr-1 pr(1 – p)x-r x = r, r+1, … r/p √ (r(1 -p)/p 2) Example: r number of successes in x trials (of a binomial experiment) What is the P(x=5) flips are necessary to get 3 heads? (λt)x P(x) = ------- e-λt Poisson x = 0, 1, 2, … λt x! Example: Events occurring over time or space What is the P(x<15) people per hour going through Mc. D’s drive through? √λt

Summary and Homework • Summary – Probability rules remain the same – Each of the six Discrete PDFs have criteria that are similar, but slightly different – Calculator has pdf and cdf functions for 3 of the 6 Discrete PDFs • Homework – pg 352 - 355: 1, 4, 5, 9, 13, 19