Math/Stat 3850 Exam 1

Transcription

1 2/21/18 Name: Math/Stat 3850 Exam 1 There are 10 questions, worth a total of 100 points. You may use R, your calculator, and any written or internet resources on this test, although you are not allowed to ask for help from anyone except Dr. Clair. (10) 1. Use R to calculate the sum of the square roots of each number from 1 to That is, compute Write your answer and your R code here. Solution: sum(sqrt(1:1000)) gives (10) 2. The data set quakes is built in to R. It gives information about earthquakes near Fiji. (a) What was the largest magnitude of an earthquake in this dataset? (b) What was the average depth of earthquakes in this dataset? (c) Make a histogram of stations and describe it. Solution: a. 6.4; b ; c. A large number of quakes have a small number of stations report them, and very few quakes have a large number of stations. The distribution is right skew.

2 Define f(x) = 1 2 { 1 2 sin(x) 0 x π 0 otherwise as shown in the picture. 0 π 2 π Let X be a random variable with pdf given by f. (10) 3. (a) What is E(X)? (b) Estimate σ(x) by eye (you don t need to be very accurate). (c) Draw on the graph the mean and a spread of one standard deviation around the mean. Solution: a. 1/2; b. It s exactly 1 2 π (10) 4. Continue with X and f(x) as above. (a) What is P (X > 0)? (b) What is P (X > π 2 )? (c) What is P (X π 4 )?

3 Solution: a. 1; b. 1/2; c. 1 2 ( )

4 (10) 5. Suppose you do an experiment where you select ten people at random and ask their birthdays. Here are three events: A : all ten people were born in February B : the first person was born in February C : the second person was born in January Which pair(s) of these events are disjoint, if any? Which pair(s) of these events are independent, if any? What is P (B A)? Solution: a. A and C are disjoint. b. B and C are independent. c. P (B A) = 1. (10) 6. Heights (in inches) of young women follow approximately the Norm(64, 2.7) distribution. Heights of men the same age follow the Norm(69.3, 2.8) distribution. What percent of young women are taller than the mean height of young men? Solution: 1-pnorm(69.3,64,2.7) gives

5 (10) 7. Let X be the value of rolling one six-sided die. (a) What is E(X)? (b) What is E(X 2 )? (c) What is E(X 3 )? Solution: a. 3.5; b ; c (10) 8. A 2017 study by British researchers including celebrity mathematician Rachel Riley, determined that the chance of finding love on a given day in the U.K. is 1/562. What is the probability that a single Brit finds love within 730 days (= two years)? Solution: About 72.7%. Each day is a Bernoulli trial with p = 1/562. Let X be number of loveless days before success. Then X is geometric and we want P (X 729). pgeom(729,1/562) gives Or, you can work out the probability of failing for 730 days in a row: 1 ( ) This can be done as a binomial experiment with 730 days, and X the number of days on which you find love. Then P (X > 0) = 1 P (X = 0) is given by 1-dbinom(0,730,1/562) Thinking of the situation as a continuous Poisson process is also reasonable, with rate λ = 1/562. Then pexp(730,1/562) gives , almost exactly the same as the discrete approach. If you really want to use a Poisson variable, then the expected number of successes is 730/562 and X Pois(730/562) gives the number of days you find love during the two year time interval. You want X > 0, so compute 1-dpois(0,730/562).

6 (10) 9. Let X Poisson(3) and Y Poisson(2) be two independent random variables. Use simulation to compute P (X > Y ). Write your answer and your R code below. Solution: X <- rpois(10000,3); Y <- rpois(10000,2); mean(x > Y) gives approximately (10) 10. Let X Uniform( 1, 1) and Y Uniform( 1, 1) be two independent random variables. Then (X, Y ) is a random point on the square shown below. The probability that (X, Y ) is inside the unit circle is given by P (X 2 + Y 2 < 1) = Area of circle Area of square = π Use R to generate 100,000 values of X and Y, then estimate P (X 2 + Y 2 < 1). Multiply your result by 4 to get an approximation to π. Write your code here: Solution: > X <- runif(100000,-1,1) > Y <- runif(100000,-1,1) > pioverfour <- mean(x^2 + Y^2 < 1) > 4*pioverfour [1]