STA 032 Spring 2016 Homework 5 - Book Portion - Due Friday, May 13th Reminder: Book homework and R homework are to be turned in to separate piles (a) Find the probability the first item sold for more than the second item. (b) Find the probability that both items sold for over 400 gold pieces total. (c) Find the probability that the average of the two items sold for between 220 and 250 gold pieces. Book Homework (d) Find the 30th percentile for the average price of the two items. 1. A doctor knows from experience that the probability a patient shows side effects on a particular drug is 0.10. Assume that probability does not change, and that patients are independent. (e) Find the 60th percentile for the total of the two items. (f) Find the value of the total price for both where only 10% of the time, both items will sell for more than that value. (a) How many patients on average should he expect to give the drug before one of them has a side effect? (b) What is the probability that the 4th patient he gives the drug to is the first to have a side effect? (c) What is the probability that the 2nd patient to have a side effect occurs between the 4th, 5th, and 6th patient to receive the drug? (d) What is the probability that the 3nd patient to have a side effect occurs exactly on the 6th patient to receive the drug? 5. The average amount of time until a car accident on a particular 60 mile stretch of road is 60 minutes. Assume (unreasonably) that car accidents are independent, and that two accidents cannot occur at the same time. (a) What is the probability of a car accident occurring in the two hours? (b) What is the probability of a car accident occurring between 45 and 75 minutes? 2. At a particular manufacturing plant, the probability that a created item does not meet specifications is 0.04. The assembly will stop for recalibration if it detects 2 items that do not meet specifications. Assume items are independent, and the probability of not meeting specifications does not change. (c) What is the variance of the time until a car accident occurs? (d) If a car accident has not happened in 3 hours, what is the probability it will happen in the next two hours? (a) Find the expected number of items produced before the machine needs recalibration, and the standard deviation. (b) What is the probability the machine stops on exactly the 8th item? (c) What is the probability the machine stops after the 4th item? (d) What is the probability the first item that does not meet specifications is the 4th item? 6. In a particular state, a major flood happens once every 3 years on average. If you can assume floods are independent, and that two floods cannot happen simultaneously, find the following: (a) The probability that a flood happens within the first 5 years. (b) The probability that a flood happens after the first 2 years. (c) If there has been no flood for 4 years, what is the probability there is a flood in the next 2 years? 3. Assume that IQ is normally distributed, with mean 100 and standard deviation 15. (d) The expected time until a flood happens, and the standard deviation. (a) What is the probability that a randomly selected persons IQ is over 120? (b) Find the values of Q1 , Q2 , and Q3 for IQ. (c) Find the values of lower = Q1 1.5(Q3 Q1 ) and upper = Q3 + 1.5(Q3 Q1 ) for IQ. Recall that these were cutoffs for outliers. (d) Find the probability of an outlier for IQ for a single person based on your values for (c). (e) If we randomly selected 10 people, what is the probability their average IQ is over 105? (e) Find the 50th percentile of the time until a flood. 7. The wait time at a popular fast food restaurant is uniformly distributed between 1 and 15 minutes on average. (a) Find the probability that you wait less than 6 minutes. (b) Find the probability that you wait between 5 and 10 minutes. (c) What is the expected wait time, and the variance of wait time? 4. Assume that the selling price of a particular item (call it I1 ) is normally distributed, with mean 200 gold coins, and standard deviation 50 coins. Assume the selling price of another item is also normally distributed, with mean 230 gold coins, and standard deviation 10 gold coins (call it I2 ). Assume these distributions are independent. (d) If you have waited 5 minutes already, what is the probability you wait an additional 5 minutes? 8. The total amount of tips a waitress receives is uniformly distributed between $1 and $20 in an hour on average. 1 (a) Find the probability the waitress receives between $5 and $10. III. Consider the function rnorm. This simulates a normal random variable, for example the following code will generate 1000 normal random variables with mean 5, standard deviation 6: (b) What is the expected tip amount the waitress will receive, and the standard deviation? (c) Find the following probabilities: P r(X < X ), P r(X > X ) where X =the amount of tips a waitress receives. X = rnorm(1000, mean = 5, sd = 6) (a) Generate 10000 values of a normal random variable with mean 2, standard deviation 1, and call this vector X. Generate 10000 values of a normal random variable with mean -2, standard deviation 5, and call this vector Y . Report back the mean and standard deviation of both vectors (using the functions in R). (d) Find the following probabilities: P r(X < X X ), P r(X > X + X ) where X =the amount of tips a waitress receives. (e) What is the median amount of tips a waitress will receive? (b) Create a vector W by adding the two vectors from (a) together, and find the mean and standard deviation W (using R). R Homework I. The goal of this problem is to simulate a Geometric random variable. Assume in a computer game Stardew Valley, there is a 10% chance it rains on any given day. (c) Find the probability that W is larger than 3. (d) We know what the answers to (b), (c) should be exactly. Calculate the error for all the values found (the error is what R found subtract by what we know it should be). (a) Use R to simulate generating days up to and including the first time it rains. Return based on one (random) run of your simulation the number of days it took up to and including the first rainy day. (b) Repeat (a) 20000 times, so that you should have 20000 values of how many days it took up to and including the first rainy day. Do not print out these values!!! Plot a histogram of these values. (c) What can you say about the distribution of this particular Geometric random variable based on (b)? Describe it. (d) Find the expected value and standard deviation of your vector in (b). (e) Find the probability that it took 15 days until the first rainy day. (f) Find the probability that if it took more than 10 days, it would take an additional 5 days to observe the first rainy day. II. The goal of this problem is to simulate a Negative Binomial random variable. Consider that 40% of people wear glasses for near-sightedness. (a) Use R to simulate selecting random people until 5 are selected wear glasses. Return one the number of people it took until 5 wore glasses. (b) Repeat (a) 20000 times, so that you should have 20000 values of how many people were were selected until 5 wore glasses. Do not print out these values!!! Plot a histogram of these values. (c) Find the median of the vector from (b). (d) Find the mean and standard deviation of your vector from (b). (e) Find the probability we had to sample more than 10 people before we found 5 that needed glasses. (f) Find the probability that if it takes more than 10 people, it will an additional 5 to find 5 that wore glasses. 2