Worksheet 4 - NYU Polytechnic School of Engineering

NYU Polytechnic School of Engineering
MA 3022 SPRING 2014
WORKSHEET IV
Print Name:
Signature:
Section:
Instructor: Dr. Manocha
Date:
ID #:
Directions: Complete all questions clearly and neatly. You must show all work to have
credit. Unclear work will not be graded.
Problem Possible
1
12
2
12
3
12
4
12
5
12
6
12
7
14
8
14
Total
100
Points
Your signature:
(1)(12 points)
Let Y = X1 + X2 + ... + X15 be the sum of a random sample of size 15 from distribution
whose p.d.f. is f (x) = (3/2)x2 , −1 < x < 1. Using the p.d.f of Y , we find that P (−0.3 ≤
Y ≤ 1.5) = 0.22788. Use the central limit theorem to approximate this probability.
Your signature:
(2)(12 points)
Let X equal the weight in grams of a miniature candy bar. Assume that µ = E(X) = 24.43
¯ be the sample mean of a random sample of n = 30 candy
and σ 2 = V ar(X) = 2.20. Let X
bars. Find
¯
(a) E(X)?
¯
(b) V ar(X)?
¯ ≤ 24.82), approximately.
(c) P (24.17 ≤ X
Your signature:
(3)(12 points)
Let X1 , X2 , ..., X30 be a random sample of size 30 from a Poisson distribution with a mean of
2/3. Approximate.
(a) P (15 <
30
P
Xi ≤ 22)
i=1
(b) P (21 ≤
30
P
i=1
Xi < 27)
Your signature:
(4) (12 points)
A die is rolled 24 independent times. Let Y be the sum of the 24 resulting values. Recalling
that Y is a random variable of discrete type, approximate
(a) P (Y ≥ 86).
(b) P (Y < 86)
(c) P (70 < Y ≤ 86).
Your signature:
(5) (12 points)
An insurance company has 10,000 automobile policyholders. The expected yearly claim per
policyholder is $240, with a standard deviation of $800. Approximate the probability that the
total yearly claim exceeds $2.7 million.
Your signature:
(6) (12 points)
Events occur according to a Poisson process with rate λ = 3 per hour.
(a) What is the probability that no events occur between times 8 and 10 in the morning?
(b) What is the expected value of a number of events that occur between times 8 and 10 in
the morning?
(c) What is the expected time of occurrence of the fifth event after 2 P.M.?
Your signature:
(7)(14 points)
Customers arrive at a certain retail establishment according to a Poisson process with rate λ
per hour. Suppose that two customers arrive during first hour . Find the probability that
(a) both arrived in the first 20 minutes;
(b) at least one arrived in the first 30 minutes.
Your signature:
(8)(14 points)
If X is b(100, 0.1), find the approximate value of P (12 ≤ X ≤ 14), using
(a) The normal approximation.
(b) The Poisson approximation.
(c) The binomial.