(a) Show that the above situation satisfies all four criteria for the Binomial probability distribution.

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1 Homework 3 for Statistics Binomial and Poisson Distributions PRINT NAME: Problem 1: A new drug named CURAIDS that is 60% effective in extending the average life of an AIDS patient by twenty years. Five randomly selected AIDS patients from Africa are treated with this new drug. Answer the following questions based on the above information. (a) Show that the above situation satisfies all four criteria for the Binomial probability distribution. (b) (10 points) Fill in the probabilities in the following table. Show your calculations. x P(x) 1 P a g e

2 (c) What is the probability that no more than 4 patients are cured? Use results from part (b), do not do the calculations again. (d) Find the probability that more than 2 patients or less than or equal to 5 patients are cured. Use results from part (b), do not do the calculations again. (e) Find the probability that at least 4 patients are cured. Use results from part (b), do not do the calculations again. (f) Find probability that less than 2 or more than 3 patients are cured. Use results from part (b), do not do the calculations again. 2 P a g e

3 Problem 2: Your local furniture shop is giving this special credit offer where no credit check and no down payment for purchases are required. From the past experience, the store owner knows that the probability that someone who takes this offer will not honor the easy credit extension is 10%. On particular day, the furniture shop extended credit to 10 new customers. Find the following probabilities: a) Show that this forms a Binomial Distribution b) Find the probability that at least one customer will not honor this credit offer c) Find the probability that at least 8 customers will honor this credit offer d) Find the mean and standard deviation for number of customers who WILL NOT honor this credit offer e) Find the mean and standard deviation for number of customers who WILL honor this credit offer 3 P a g e

4 Problem 3: S&P 500 is around 2000 now and has a 60% chance of gaining 3% or more over the next year; assume that this probability remains constant over the next 5 years. a) Show that this is a Binomial Distribution b) What is the probability that S&P 500 will gain 3% or more at least 3 out of the next 5 years. c) What is the expected return for S&P 500 d) What is the standard deviation for S&P 500 e) What is the variance of S&P P a g e

5 Problem 4: Airlines companies always try to overbook if they can because some people will almost always not show up for their flight on time or will completely miss it. A beautiful small town airport named Monterey Airport flies a small flight every day between Monterey and San Francisco that carries at most 20 people. However, they over booked the flight by two people. Probability that a person would miss a flight is about 15%. Please answer the following: a) Show that this scenario forms a Binomial Distribution. b) What is probability that the flight will have sufficient seats for all the passengers who show up? c) What is the probability that they will not have enough seats for the passengers who arrive for the flight on time? 5 P a g e

6 Problem 5: According to Experian, U.S. smartphone owners aged 18 to 24 send 2,022 texts per month on average 67 texts on a daily basis and receive another 1,831 or 61 messages per day (March, 2012). Assume that arrival of text messages is a random event and has a mean of 61 messages per day. Answer the following: a) Show that this scenario matches the conditions of Poisson Distribution b) Find the probability that on particular day someone receives 75 messages Problem 6: At your local restaurant for the lunch period (11:00 a.m. to 2 p.m.) on the average 80 people show up. The arrival of customers is a random event. a) Show that this scenario matches the conditions of Poisson Distribution b) Find the probability that on particular hour 40 customers show up for lunch 6 P a g e

7 Problem 7: Texting While Driving could cause serious accidents and/or deaths. In the Bay Area, on average 100 such accidents happen per year that result in taking the Texter to the emergency room. a) Show that this scenario matches the conditions of Poisson Distribution b) Find the probability that on a particular day at most 2 such emergency room visits will happen 7 P a g e

Section 6.1 Discrete Random variables Probability Distribution

Section 6.1 Discrete Random variables Probability Distribution Definitions a) Random variable is a variable whose values are determined by chance. b) Discrete Probability distribution consists of the values