1.Suppose that X and Y are independent random variables, X is exponentially dis- tributed with parameter λ = 2, and Y is uniformly distributed on (1, 3).

FindP(Y <X), Derive the pdf of Z=X+Y.

2. The random variables X, Y, and Z are independent and uniformly distributed on

(0,1). Find the density function of X+Y +Z.

3 . A table is ruled with equidistant horizontal and vertical lines at distance D apart. A needle of length L, L ≤ D, is randomly thrown on the table. What is the expected number of lines crossed by the needle? What is the probability that at least one line will be crossed?

4.Compute the expected number of distinct birthdays for a class of 40 people.

5.Suppose that a random variable X satisfies E[X] = 0, E[X^2] = 1, E[X^3] = 0, E[X^4] =3. Let Y = a + bX + cX^2. Find the correlation coefficient ρ(X, Y ).

6. Twenty crows land randomly on a wire. Each crow is crowing at the nearest crow. What is the expected number of crows that are not crowed at?

7.n people come to the party, each wearing his/her own hat. By the end, everybody had too much fun and randomly picks up a hat before leaving. People i and j are said to form a matched pair if i chooses the hat belonging to j, and j chooses the hat belonging to i. Find the expected number of matched pairs.

8. A total of 2015 balls, numbered from 1 through 2015, are placed into 2015 boxes, also numbered from 1 through 2015, in such a way that ball i is equally likely to go into any of the boxes 1, 2, . . . , i. Find the expected number of empty boxes.

Subject | Mathematics |

Due By (Pacific Time) | 11/28/2014 12:00 pm |

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