Week 6, 9/24/12-9/28/12, Notes: Bernoulli, Binomial, Hypergeometric, and Poisson Random Variables 1 Monday 9/24/12 on Bernoulli and Binomial R.V.s We are now discussing discrete random variables that have names. Any time on a quiz or test that you are working such a problem, you should define COMPLETELY the r.v. As an example, suppose I was looking at the number of heads in 23 tosses of a coin that had a probability of heads of.45 in 1 toss. If X was the number of heads in this experiment, I would write X Binomial(n=23, p=.45). I have written the name of the distribution (an appropriately shorthand like Bin would be acceptable), the parameters of the distribution and the values of the parameters as based on our model. You must have all 3 things when defining your r.v. Many problems in probability involve independently repeating a random experiment and observing at each repetition whether a specified event occurs. We label the occurrence of the specified event a success and the nonoccurrence of the specified event a failure. A success could be a female child, a head from a coin flip, a 5 on a die, a defective part in a manufacturing warehouse, a green spin in roulette, etc. A success can take on a positive or negative connotation in the context of an example; it is merely the event that we are interested in. Each repetition of the random experiment is called a trial. We use p to denote the probability of a success on 1 trial. In Bernoulli Trials, p remains constant from trial to trial. Conditions for Bernoulli: The trials are independent of one another. The result of each trial is classified as a success or failure, depending on whether or not a specified event occurs respectively. The success probability and therefore the failure probability remains the same from trial to trial. An important note: If we sample from a population 1 at a time, it is Bernoulli if we sample with replacement, but it is not Bernoulli if we sample without replacement. Sometimes the Bernoulli Distribution is called an indicator function, i.e. it lets one know whether or not a specific event has occurred. Characteristics of the Bernoulli Distribution:
The support is: The expected value is: We can define the Binomial R.V. as the number of successes in n independent (redundant?) Bernoulli trials, where the probability of success in one trial is p. Characteristics of the Binomial Distribution: The support is: The expected value is: There are several approximations in this course. All 3 of them involve the Binomial in some way. These will be written in later on where appropriate. However, I give a quick summary here. We can use the Binomial to approximate the Hypergeometric of N > 20n. We can use the Poisson to approximate the Binomial if n > 100 and p <.01. We can use the Normal to approximate the Binomial if np > 5 and n(1-p) > 5. Example 6.1a In Chris Stat 225 class, 75% of the students passed (got a C or better) on Exam 1. If we were to pick a student at random and asked them whether or not they passed. If we let X represent the number of student(s) who passed, what type of random variable is this? How do you know? Additionally, write down the pmf, the expected value, and the variance for X. Example 6.1b Repeat 6.1a under the following assumption: What about if we picked 10 students with replacement and let X be the number of student(s) who passed. Example 6.2 Suppose that 95% of consumers can recognize Coke in a blind taste test. Assume consumers are independent of one another. The company randomly selects 4 consumers for a taste test. Let X be the number of consumers who recognize Coke. Write out the pmf table for X. What is the probability that X is at least 1? What is the probability that X is at most 3? Example 6.3 To test for ESP, we have 4 cards. They will be shuffled and one randomly selected each time, and you are to guess which card is selected. This is repeated 10 times. You do not have ESP. Let R be the number of times you guess a card correctly. What are the distribution
and parameter(s) of R? What is the expected value of R? Furthermore, suppose that you get certified as having ESP if you score at least an 8 on the test. What is the probability that you get certified as having ESP? 2 Wednesday 9/26/12 on the Hypergeometric R.V. and the Binomial Approximation to it Important applications are quality control and statistical estimation of population proportions. The hypergeometric r.v. the equivalent of a Binomial r.v. except that sampling is done without replacement, or put another way, the trials are dependent (no longer Bernoulli trials). As an illustration, let us revisit a poker example. Assume we have a standard 52 card deck and we are drawing five cards without replacement. Let us use our counting rules to determine the probability of 3 kings. For the sake of this problem, we are going to assume we do not care what the remaining two cards are, just that they are not kings. The answer to this problem involves combinations since we are sampling without replacement, and the sampling order does not matter (because we only care about which cards we received, not in what order we received them). So, you have to answer 3 questions. How many ways are there to get 3 kings? How many ways are there to get the remaining 2 cards? How many ways are there total to get a 5 card hand? Put these altogether for the answer of (4 3) ( 48 2 ). Little did you know, you just used the hypergeometric ( 52 5 ) distribution. Characteristics of the Hypergeometric Distribution: The support is: The expected value is: What is the difference between an Binomial r.v. and a Hypergeometric r.v.? Hint: Do NOT say N. Approximation. If X Hyp(N,n,p) and N 20n, then we can approximate the probability of X by using X* Bin(n,p) (the same n and p). Example 6.4 There are 100 identical looking 52 TVs at Best Buy in Costa Mesa, California. Let 10 of them be defective. Suppose we want to buy 8 of the aforementioned TVs (at random).
What is the probability that we don t get any defective TVs? Example 6.5a An experiment consists of shuffling a standard deck of 52 cards and then dealing a 10 card hand. Let Y denote the number of hearts in the hand. Identify the distribution of Y and give its parameter(s). Find the probability that Y is 3. Example 6.5b Suppose instead of using 1 deck, we mix together 1,000 decks. The cards are shuffled and 10 are dealt into a hand. Again, let Y denote the number of hearts in the hand. Is an approximate distribution appropriate for Y, why or why not? Find the probability that Y is 3 (if an approximation is appropriate, use that instead of the exact distribution). If you used an approximation, what is the distribution and the value of its parameter(s)? Example 6.6 Jacob is shooting a basketball at a carnival in order to win a stuffed animal for his girlfriend. On a single shot, Jacob can make a basket with probability.65. Jacob will win a small prize if he makes at least 2 out of 3 shots. Jacob pays $4 for three shots. What is the probability that Jacob will win a small prize with his first $4. What distribution and what parameter(s) are you using? What is the probability it takes Jacob $20 to win hist first small prize? 3 Friday 9/28/12 on the Poisson R.V. and its Approximation of the Binomial An important fact from Calculus is: e t = n=0 tn n!. This fact will allow one to show that the pmf for a Poisson indeed sums to one for any value of λ. The Poisson r.v. also measures number of successes (like the 3 preceding named discrete r.v.s). However, it is different from the others in the fact that it does not have a sample size (or depending on perspective, you can take the sample size to be infinite). While our 3 previous r.v.s measure number of successes in a certain number of trials, the Poisson r.v. measures number of successes per [blank]. This [blank] can be something like hours, cookies, area, volume, etc. Example in the past have included: number of chocolate chips in a cookie (or batch of cookies), number of busses per hour, number of silver loop busses per hour, number of defects per square foot, etc. Characteristics of the Poisson Distribution: The support is:
The expected value is: Approximation: If X is Bin(n,p) where n > 100 and p <.01, then X can be approximated by X* Poisson(λ = np). Example 6.7 Let us say a certain disease has a probability of occurring in 7 out of 5,000 people. Let us sample 1,000 people. Find the exact and approximate probabilities that 0 people have the disease and at most 5 people have the disease. Example 6.8 Suppose earthquakes occur in the western US with a rate of 2 per week. Let X be the number of earthquakes in the western US this week. Let Y be the number of earthquakes in the wester US this month (assume a 4 week period of time). Find the probability that X is 3 and Y is 12. Let Z be the number of weeks in a 4 week period that have a week with 3 earthquakes in the western US. Find the probability that Z is 4. Is this the same as the probability that Y is 12? Does this make sense? Additional Examples Here: Example 6.9 A store has 50 light bulbs for sale. Of these, 5 are black lights. A customer buys eight light bulbs randomly chosen from the store. Let B denote the number of black light bulbs the customer selected. Define the distribution of B. What is the probability that B is 1? What is the probability the customer gets at least one black light bulb? Example 6.10 PRP has on average 4 telephone calls per minute. Let X be the number of phone calls in the next minute. Find the probability that X is at least 3. Example 6.11 Customers arrive at the VP on 9 th Street at a rate of 10 per hour. What is the distribution of the number of customers that arrive in the first 3 hours, call this distribution Y? What is the probability that exactly 12 customers arrive in each of the first 3 hours? What is the probability that Y is 36? Example 6.12 You are interested in the Indianapolis Indians. They play 20 games in the month of August. Of their games, they win 10% of them by 2 runs or fewer. Assume each game is independent of any other game. Let G be the number of games won by the Indians by 2 or fewer runs. What is the distribution and parameter(s) of G? Wbat is the probability that G is either 2 or 3? If the Indians win 4 or more games by 2 or fewer runs in August, they will receive $20,000 bonuses. What is the probability the players receive bonuses? Given the players do not receive bonuses, what is the probability that they win exactly 3 games by 2 runs or fewer?
What is the expectation of G? What is the variance of G? Example 6.13 A girl scout troop has 100 boxes of cookies to sell. Of these 100 boxes, 60 are thin mints and 40 are Samoas. 10 boxes are randomly selected to be sold at the White County Fair. Let S be the number of boxes of Samoas selected to go to the fair. What is the distribution of S as well as the value(s) of its parameter(s)? Find the probability that S is 0. Suppose that thin mints can sell for $4 and Samoas can sell for $3.50. What is the expected value and standard deviation of the amount of money the girl scouts will make at the fair (assume that all 10 selected boxes will be sold). Example 6.14 Tom Maloney decided to hang out with friends the night before his quiz and did not study. He has no knowledge of any of the material on the quiz. The quiz consists of 5 multiple choice questions with 3 possible answers each. Let T be the number of answers that Tom correctly guesses. What is the distribution and parameter(s) of T? What is the probability that Tom gets at least a B (on our grading scale)? Example 6.15 Flaws on a used computer tape occur on the average of one flaw per 1,200 feet. Let X denote the number of flaws in a 4,800 foot roll. Name the distribution of X. What is the probability that X is at least 1?