I am having some trouble with my hw....ne help will be much obliged.
2. Suppose that 30% of the applicants for a job can program in C. The company needs to find four C programmers and sets up sequential interviews until they find the four people.
a. What is the probability at least 10 interviews are required?
b. What is the mean and standard deviation of the number of interviews required?
c. If the company wants to be at least 90% certain that they have scheduled enough interviews to find four C programmers, how many interviews do they need to schedule? (Hint: Find smallest n so that Binomdist(3,n,0.4,1) ≤ 0.1); why does this work?)
3. Let Y be the number of trials until the first success with Bernoulli trials with success probability equal to p.
a. Show P(Y > n) = (1 – p)n.
b. Show P(Y > n + m | Y > n) = P(Y > m). This property is called the memoryless property of the geometric random variable.
c. Give an intuitive explanation for the memoryless property of the geometric distribution.
d. Find an equation for n so that P(Y ≤ n) ≥ .95.
4. Suppose 20% of CSUN students use marijuana “regularly.” Suppose you ask students if they use marijuana regularly. If a student is not a user, he/she answers “No.” If he/she is a user, they answer, “No,” with probability equal to .75.
a. What is the probability of a “Yes” answer?
b. How many students, on average, do you need to ask to find a sample of five “yes” respondents?
c. What is P(a student is a user | the student says they are not)?
d. Randomized response: To help get more truthful responses, sometimes we ask the following question. If the last digit in your student ID is odd, answer the question,
“Do you use marijuana regularly?” If the last digit is even, answer the question, is the next to last digit odd? Now what is the probability of a “yes” response?
e. On average, how many yes responders do you need (to the randomized response question) to have a sample that averages at least 3 regular users? (You should calculate P(User | “yes” response).
5. Let Y be a Poisson random variable with mean λ = 2. Find:
a. P(Y = 3)
b. P(Y < 3)
c. P(Y ≤ 3)
d. P(Y ≥ 3)
e. P(Y > 3)
6. Show that if X is Poisson with mean λ, and Y is Poisson with mean μ, and X and Y are independent, then X + Y is Poisson with mean (λ + μ). For example,
P(X + Y = 3) = P(X = 0, Y = 3) + P(X = 1, Y = 2) +P(X = 2, Y = 1) + P(X = 3, Y = 0). You will have to sum terms of the form [IMG]file:///C:/Users/Shaun/AppData/Local/Temp/msohtml1/01/clip_image002.gif[/IMG]. What these sum to might be more obvious if you multiply and divide by 3! and recall the binomial theorem. Also, keep an eye on the answer you hope to show.
7. Use the result of problem 6, whether you could prove it or not. The North entrance to a parking lot averages 3 entries per hour. The South entrance averages 2 entries per hour. Let Y be all of the entries to this parking lot from 9 AM to 11 AM.
a. Find P(Y ≥ 8)
b. Find the mean and standard deviation of Y.