Math 662- Exam I Name:__________________

November 10, 1998 Student #:________________

Must show all work for full credit!!!

I pledge that I have not violated the NJIT code of honor____________________________

1. A grain loader can be set to discharge grain in amounts that are normally distributed with mean m bushels and a standard deviation equal to 25.7 bushels. If a company wishes to use the loader to fill containers that hold 2000 bushels of grain and wants to overfill only one container in 100, at what value of m should the company set the loader?

(15 pts)

2(a) Let X₁, X₂,…,X_n be mutually independent identically distributed with density f(x) > 0

defined on the real line. Find the distribution of the number of variables

X_i, i = 1,2,…,n that lie in the interval (a, b).

(15 pts)

(b) Define the mutual independence of k events A₁, A₂, …, A_k. If these A’s are

mutually independent what can you say about the independence of A₁, A₂, A₃, not

A₄, …,(all negations), and not A_k. Explain your answer with logical details.

(15 pts)

3. An automobile insurance company classifies each driver as a good risk (A1), a medium risk (A2), or a poor risk (A3). Of those currently insured, 30% are good risks,

50% are medium risks and 20% are poor risks. In any given year, the probability that

a driver will have at least one citation is 0.1 for a good risk, 0.3 for a medium risk,

and 0.5 for a poor risk. If a randomly chosen driver insured by this company has at

least one citation during the next year then what is the probability that the driver was

actually a good risk?

(15pts)

4. The joint probability density of X and Y is given by

f_x,y(x, y) = 3(x + y)I_(0,1)(x) I_(0,1)(y) I_(0,1)(x+y)

(a) Find Marginal density of X

(10 pts)

(b) P(Y < 0.5/ X = 0.3)

(10 pts)

(c) P(Y< 0.5/ X < 0.3)

(10pts)

(d) The correlation r_x,y

(10 pts)