Math 662- Exam I    Name:__________________

October 23, 2000                                               Student #:__________________

 

Must show all work for full credit!!!

 

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

 

 

  1. If A1, A2,…are sets such that Ak É Ak+1,…, k = 1, 2, …and lim k→∞ Ak is defined as the intersection  A1 Ç A2 Ç A3 Ç …. Find lim k→∞ Ak, where Ak =

(10 pts)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  1. Find the 36th percentile of the distribution having probability density 

f(x) =

 (12 pts)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  1. The number of customers who visit a car dealer's showroom on a Saturday morning is a random variable with m = 18 and s = 2.5. With what probability can we assert that there will be between 8 and 28 customers?

(12 pts)

 

 

 

 

 

 

 

 

  1. 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?

(12pts)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  1. A deck of playing cards contains 52 cards from which the 13 cards are selected at random and without replacement. Compute the probability of being dealt a 13-card bridge hand consisting of

 

(a) 6 spades, 4 hearts, 2 diamonds, and 1 club.

(8 pts)

 

 

 

 

 

 

(b) 13 cards of the same suite.

(8 pts)

 

 

 

  1. Let  X  be uniformly distributed on the interval (-1, 1). Compute the probability density function of Y = 1 - X2.

(15pts)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  1. Given below is the cumulative distribution function of the random variable

F(x) =

Compute              (i)  P(-1 £ X < 1)

                     (5 pts)

 

 

 

 

 

 

 

 

                            (ii) P( X > 0 / -1 < X < ⅛)

                                (6 pts)

 

 

 

 

 

 

  1. Let E= , r = 1, 2, 3,….  Find the series representation for the moment generating function of X. Sum this series and find a closed form expression for it.

(12 pts)