Introduction to Probability and Statistics

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

Binomial Random Variable The coin-tossing experimentis a simple example of a binomial random variable. Toss a fair coin n = 3 times and record x = number of heads.

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 • The experiment consists ofn identical trials. • Each trial results in one of two outcomes, success (S) or failure (F). • Probability of success on a single trial is p and remains constant from trial to trial. The probability of failure is q = 1 – p. • Trials are independent. • Random variable x, the number of successes in n trials. 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? m m m m m m Yes, 3 trials are independent with same probability of getting a green.

m m m m m m 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.

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

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

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

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 pof 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 pof success on a given trial, the measures of center and spread are:

p = x = n = 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? hit .8 # of hits 5

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

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

Cumulative Probability 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)

Key Concepts I. The Binomial Random Variable 1. Five characteristics: the experiment consistsof n identical trials; each resulting in either success S or failure F; probability of success is pand 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 • probability that exactly 8 households own dogs? • probability that at most 3 households own dogs? • probability that more than 10 own dogs? • the mean, variance and standard deviation of x, the number of households that own dogs.

p = x = n = 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 own dog .4 # 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

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

Introduction to Probability and Statistics