X P(X=x) E(X)= V(X)= S.D(X)= X P(X=x) E(X)= V(X)= S.D(X)=

Transcription

1 1. X P(X=x) E(X)= V(X)= S.D(X)= X P(X=x) E(X)= V(X)= S.D(X)= 2. A day trader buys an option on a stock that will return a $100 profit if the stock goes up today and loses $200 if the stock goes down. If the trader thinks that there is a 75% chance that the stock will go up. a) Create a probability model b) What is the expected value of the option and its standard deviation? 3. You draw a card from a deck of cards. If you get a red card, you win nothing. If you get a spade you win $5. For any club you win $10 plus an extra $20 for the ace of clubs. a) Create a probability model: b) Find the expected amount you will win at this game? 4. You play two games with the same opponent. The probability you win the first game is 0.4. If you win the first game, the probability that you also win the second game is 0.2. if you lose the first game, the probability that you the second is 0.3. a) Are the two games independent? b) What s the probability that you lose both games? c) What s the probability that you win both games? d) Let random variable X denote the number of games you win. Create a model for X. e) What are the expected value and standard deviation?

2 Mean S.D X 10 2 Y Random Variables: Given two independent random variables with means and standard deviations as shown below: Find the mean and standard deviation for the following: Linear Combination of RVS Mean Standard Deviation a) 3X b) Y+7 c) -0.5Y d) X-Y e) X+Y f) 4X-10Y 6. A grocery supplier believes that in a dozen eggs, the mean number of broken ones is 0.6 with a standard deviation of 0.5 eggs. You buy 3 dozen eggs without checking them. a) How many broken eggs do you expect to get? b) What s the standard deviation? c) What assumption did you have to make about the eggs in order to answer the above questions? 7. Suppose the time is takes a customer to get and pay for seats at the ticket window of a baseball park is a random variable with a mean of 100 seconds and a standard deviation of 50 seconds. When you get there you only find two people in line in front of you. a) How long do you expect to wait for your turn to get tickets? b) What s the standard deviation of your wait time? c)what assumption are you making about the two customers in finding the standard deviation?

3 Binomial (1-5) & Poisson (6-9) 1.A machine that produces stampings for automobile engines in malfunctioning and producing 10% defectives. The defective and non-defective stampings proceed from the machine in a random manner. If the next five stampings are tested, find the probability that: a) three of them are defective b) at least two of them are defective c) none of them are defective 2.A basketball player has made 80% of his foul shots during the season. Assuming the shots are independent, find the probability that in tonight s game he, a) Misses for the first time on his fifth attempt. b) Makes his first basket on his fourth shot. 3. About 8% of males are colorblind. A researcher needs some colorblind subjects for an experiment and begins checking potential subjects. a) What s the probability that she won t find anyone colorblind among the first four men she checks? b) What s the probability that the first colorblind man found will be the sixth one checked? d) What s the probability that she finds someone who is colorblind before checking the 10 th man? 4. An Olympic archer is able to hit the bull s-eye 80% of the time. Assume each shot is independent of the others. If she shoots 6 arrows, what s the probability of each of the following results? a) Her first bull s-eye comes on the third arrow. b) She misses the bull s-eye at least once. c) Her first bull s-eye comes on the fourth or the fifth arrow.

4 d) She gets at least 4 bull s-eyes. e) What is the expected number of bull s-eyes she may get? 5. Suppose individuals with a certain gene have a 0.70 probability of eventually contracting a certain disease. If 12 individuals with the gene participate in a lifetime study, a) What is the probability that at most 4 people contract the disease (i.e. P(X 4)) b) How many people do you expect would contract the disease? c) What is the standard deviation of the number of people who contract the disease? 6. Births in a hospital occur randomly at an average rate of 1.8 births per hour. a) What is the probability of observing 4 births in a given hour at the hospital? b) what is the probability of observing more than 3 births in a given hour? c) What is the standard deviation for the number of births? d) What is the probability that at most 2 births will occur in a given hour? 7. Consider a computer system with Poisson job-arrival stream at an average of 2 per minute. Determine the probability that in any one-minute interval there will be a) 0 jobs; b) exactly 2 jobs; c) at most 3 arrivals. 8. A small life insurance company has determined that on the average it receives 6 death claims per day. a) Find the probability that the company receives at least seven death claims on a randomly selected day. b) Find the probability that the company receives exactly 4 death claims per day. c) Find the probability that the company receives at least 3 death claims per day.

5 9. The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 7. a) Find the probability that less than three accidents will occur next month on this stretch of road. b) Find the probability of observing exactly three accidents on this stretch of road next month. c) Find the probability that the next two months will both result in four accidents each occurring on this stretch of road. Normal and Uniform distribution (10-18) (Review of z-scores)[10-11] 10. The average score on a Stats midterm was 75 points with an SD of 5 points. If Gregor s z-score is -2, how many points did he score? 11. People with z-scores greater than 2.5 on an IQ test are considered as geniuses. If IQ tests have a score of a mean of 100 and SD of 16 points, what is the cut off score for a genius to prove himself as one? 12. Find the percent of a standard Normal model found in the given region. a) P( Z > 1.82) b) P( Z <.67) c) P( 2.87 Z 0.5) 13. Some IQ tests are standardized to a Normal model, with a mean of 115 and a standard deviation of 10 what is the probability that a randomly selected person who took the test has an IQ score? a) Over 95? A B C D b) Within 120 and 145? A B C D c) Under 100? A B C D Based on the previous question, if a randomly selected person scored among a) The top 5%, what could be his lowest score? A B C D b) The lowest 40%, what would be his maximum score? A B C c) Middle 60 %, what would be the range of the scores in which his may lie? A B C D. 25

6 15. While only 5% of babies have learned to walk by the age of 10 months, 75 % are walking by 13 months of age. If the age at which babies develop the ability to walk can be described by a Normal model find mean and standard deviation? 16. Based on a long term investigation, researchers have suggested that the acidity (ph) of rainfall in the Shenandoah mountains can be described by the normal model N(4.9,0.6) ; The lower the ph, more acidic the rain. What is the ph level for the most acidic 20% of all storms and that of the least acidic 5%? 17. Let X have a continuous uniform [0,9] distribution. a) P(X<5)= A B C D b) P(X>3)= A B C D c) E(X)= A. 4.5 B. 3.6 C. 6.6 D d) σ(x)= A B C D According to Brighton Webs Ltd., a British co. that specializes in data analysis, the arrival time of requests to a web server within each hour can be modeled by a uniform distribution. Specifically, the number of seconds x from the start of the hour that the request is made is uniformly distributed between 0 and 3600 seconds. a) Find the probability that a request is made to a web server sometime during the last 15 minutes of the hour. b) Find the probability that a request was made anytime within 10 to 40 minutes of the hour? c) What is the expected time in seconds for the request to be made within the hour? d) What is the variance in seconds?

7 Normal approximation to binomial 18. Uni airlines sell ticket more than the seats to prevent loss in case passengers do not show up. Suppose airlines sells 150 tickets the probability of any passenger boarding the flight is 0.9. There are total of 140 seats in the plane. i) What is the probability that a passenger (doctor) do not get a seat even after buying the ticket (overcrowding)? a) Calculate exact probability b) Calculate approximate probability using normal approximation and state the conditions needed for approximation. ii) What is the probability that 100 passenger shows up? a) Calculate exact probability. b) Calculate approximate probability using normal approximation.