Question

1. Suppose that random variables X and Y are independent and have the following properties: E(X) = 5, Var(X) = 2, E(Y ) = −2, E(Y 2) = 7. Compute the following. (a) E(X + Y ). (b) Var(2X − 3Y ) (c) E(X2 + 5) (d) The standard deviation of Y

. 2. Consider the following data set: �x = {90, 88, 93, 87, 85, 95, 92} (a) Compute x¯. (b) Compute the standard deviation of this set.

3. Suppose that there is a classroom with 20 students. Each of those students independently flips a fair coin. (a) What is the probability that exactly 13 students will flip HEADs and 7 students will flip TAILs? (Do not use a table) (b) Use the binomial table for this question. What is the probability that at least 13 students will flip HEADs? (c) Is the outcome of 13 HEADs more than one standard deviation above expected outcome?

4. Suppose that there is a classroom of 200 students. Among those students, 50 are from the West, 75 are from the East, 25 are from the North, and 50 are from the South. Suppose I decide to form a committee of 8 students. (a) What is the probability that there will be exactly 3 students from the South? (b) What is the expected number of students from the South? (c) What is the standard deviation for the number of students from the South?

5. Suppose that in a city of 10,000 people, there are 4,000 who like football and 6,000 who do not. Suppose that we conduct a poll of 16 citizens. What is the probability that at least half of those polled like football? (Use Binomial approximation to find a decimal answer.) 1

6. Suppose that a random variable X is an Exponential Random Variable with parameter β = 3. (a) What is E(X)? (b) Compute P(X > 2). (c) Compute P(X > 5 | X > 3).

7. An assembly line at a plant produces exactly 10000 widgets a day. Suppose that approximately 1 out of every 2000 fails a standards test and is thrown out. What is the probability that there will be 10 or more widgets thrown out on a given day?

8. There are 100 green balloons and 150 red balloons in a bag. Suppose we extract 10 balloons from the bag. (a) What is the exact probability that five of the balloons will be green? (b) Use Binomial Approximation to find the probability that exactly five of the balloons will be green. (c) Use Binomial Approximation to find the probability that no more than four of the balloons are green.

9. Suppose that X is a continuous random variable and is uniformly distributed over the interval [10, 20]. (a) What is P(X > 13)? (b) What is P(X > 13 | X < 17)?

10. The heights of women in the US are roughly normally distributed with a mean of 54 inches and a standard deviation of 3 inches. (a) Suppose there will be a new student attending class next week. What is the probability that she will be taller than 60 inches? (b) Let X represent the height of a new student. For what value k is it true that P(X > k) = 0.01?

11. Historically, the grade distribution for a certain test has been normally distributed with a mean of 80 and a standard deviation of 5. Sally is taking the exam next week and wants to know the probability that she will score at least a 90. If she assumes her outcome will be like those of students who have taken the class in the past, what is this probability?

12. Suppose that Z is a standard normal random variable. (a) Compute P(Z > 1.5). (b) What is the probability that Z is larger than 1.5 given that it is positive?

13. Suppose that the number of bees in a flower patch is roughly 5 per square meter. A gardener has a flower patch that is 3 m by 1 m. (a) What is the probability that there will be between 10 and 20 bees in the patch (inclusive of 10 and 20)? (b) What is the largest value k such that P(X ≤ k) = 0.05?

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Answer 1

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Criver E (N=51 7 E(X) = 5, var x=2, Elya-2, ECV LEC+2) = vaocx) + LECXU? = 2 + (1 - 27 Narcyz = E(Y2, - fylla 7-4 = 3 = G also given x 4 y are ind 7 COUCX,Y/=0 E (x + y) = ECXIT Ely = 5+ (-2) = 3 va812x-3y)= V001287+ Vap(-37) + lovi 27,- 3 YA - 4. Var (X2 + 9.19514) - 6.100(X,) b ☺ = 4.2+ 9.3 - 640 (since XA y 900 ind. = 8+27 = 35- E (X²+5) = E( X45 Sicce El Const. )= const. = 27+5 = 32 S.DLYIs svartur = ss ( colorleted above) leb