Introduction to Probability and Statistics Chapter 5 Discrete Distributions

Discrete Random Variables Discrete random variables take on only a finite or countable many of values. Number of heads in 1000 trials of coin tossing Number of cars that enter UNI in a certain day Number of heads in 1000 trials of coin tossing Number of cars that enter UNI in a certain day

Binomial Random Variable coin-tossing experiment binomial random variable.The coin-tossing experiment is a simple example of a binomial random variable. Toss a fair coin n = 3 times and record x = number of heads. xp(x)p(x) 01/8 13/8 2 31/8

Example Toss a coin 10 times For each single trial, probability of getting a head is 0.4 Let x denote the number of heads

The Binomial Experiment n identical trials. 1.The experiment consists of n identical trials. one of two outcomes 2.Each trial results in one of two outcomes, success (S) or failure (F). remains constant 3.Probability of success on a single trial is p and remains constant from trial to trial. The probability of failure is q = 1 – p. independent 4.Trials are independent. x, the number of successes in n trials. 5.Random variable x, the number of successes in n trials. x – Binomial random variable with parameters n and p x – Binomial random variable with parameters n and p

Binomial or Not? A box contains 4 green M&Ms and 5 red ones Take out 3 with replacement x denotes number of greens Is x binomial? Yes, 3 trials are independent with same probability of getting a green. m m mm mm

Binomial or Not? A box contains 4 green M&Ms and 5 red ones Take out 3 without replacement x denotes number of greens Is x binomial? NO, when we take out the second M&M, the probability of getting a green depends on color of the first. 3 trials are dependent. m m mm mm

Binomial or Not? Very few real life applications satisfy these requirements exactly. Select 10 people from the U.S. population, and suppose that 15% of the population has the Alzheimer’s gene. For the first person, p = P(gene) =.15 For the second person, p  P(gene) =.15, even though one person has been removed from the population… For the tenth person, p  P(gene) =.15 Yes, independent trials with the same probability of success

Binomial Random Variable Example:Example: A geneticist samples 10 people and x counts the number who have a gene linked to Alzheimer’s disease. Success:Success: Failure:Failure: Number ofNumber of trials: trials: Probability of SuccessProbability of Success Has gene Doesn’t have gene n = 10 p = P(has gene) = 0.15 Rule of Thumb: Sample size n; Population size N; If n/N <.05, the experiment is Binomial.

Example Toss a coin 10 times For each single trial, probability of getting a head is 0.4 Let x denote the number of heads Find probability of getting exactly 3 heads. i.e. P(x=3). Find probability distribution of x

Solution Simple events: Event A: {strings with exactly 3 H’s}; Probability of getting a given string in A: Probability of event A. i.e. P(x=3) Number of strings in A Strings of H’s and T’s with length 10 HTTTHTHTTT TTHHTTTTHT… HTTTHTHTTT

A General Example Toss a coin n times; For each single trial, probability of getting a head is p; Let x denote the number of heads; Find the probability of getting exactly k heads. i.e. P(x=k) Find probability distribution of x.

Binomial Probability Distribution For a binomial experiment with n trials and probability p of success on a given trial, the probability of k successes in n trials is

Binomial Mean, Variance and Standard Deviation For a binomial experiment with n trials and probability p of success on a given trial, the measures of center and spread are:

n = p =x = success =Example A marksman hits a target 80% of the time. He fires 5 shots at the target. What is the probability that exactly 3 shots hit the target? 5.8hit# of hits

Example What is the probability that more than 3 shots hit the target?

Example x = number of hits. What are the mean and standard deviation for x? (n=5,p=.8) 

Cumulative Probability cumulative probability tables You can use the cumulative probability tables to find probabilities for selected binomial distributions. Binomial cumulative probability: P(x  k) = P(x = 0) +…+ P(x = k) Binomial cumulative probability: P(x  k) = P(x = 0) +…+ P(x = k)

Key Concepts I. The Binomial Random Variable 1. Five characteristics: the experiment consists of n identical trials; each resulting in either success S or failure F; probability of success is p and remains constant; all trials are independent; x is the number of successes in n trials. 2. Calculating binomial probabilities a. Formula: b. Cumulative binomial probability P(x  k). 3. Mean of the binomial random variable: 4. Variance and standard deviation:

Example According to the Humane Society of the United States, there are approximately 40% of U.S. households own dogs. Suppose 15 households are selected at random. Find 1.probability that exactly 8 households own dogs? 2.probability that at most 3 households own dogs? 3.probability that more than 10 own dogs? 4.the mean, variance and standard deviation of x, the number of households that own dogs.

n = p =x = success =Example According to the Humane Society of the United States, there are approximately 40% of U.S. households own dogs. Suppose 15 households are selected at random. What is probability that exactly 8 households own dogs? 15.4own dog# households that own dog

Example What is the probability that at most 3 households own dogs?

Example What are the mean, variance and standard deviation of random variable x? (n=15, p=.4)

Binomial Probability Probability distribution for Binomial random variable x with n=15, p=0.4

Example 1.What are the mean, variance and standard deviation of random variable x? 2.Calculate interval within 2 standard deviations of mean. What values fall into this interval? 3.Find the probability that x fall into this interval.