 ## practice problems for final exam

### 1.(a) How many ways are there to divide 5 different cakes and 5 identical cookies between 2 peoplesothatthefirstpersongetsexactly3 cakes.      概率导论final exam代写

(b) How many ways are there to divide 5 different cakes and 5 identical cookies between 2 people so that the each person gets exactly 5 items.

2.There are three The first contains 2 red balls and 5 blue balls, the second contains 3 red balls and 4 blue balls and the third contains 4 red balls and 3 blue balls. An urn ischosen at random and then balls are drawn without replacement.

(a)Find the conditional probability that the third urn was chosen given that the first three draws resulted in 2 red and 1 blue ball.

(b)Findthe conditional probability that the fourth ball will be blue given that the first three draws resulted in 2 red and 1 blue ball.

3.(a) 3 people get 3 cards each from the standard 52 card  Find the probabilitythat at least one person has only red cards.

(b) 20 tables have 3 palyers and one standard 52 card deck per table. At each table each player is given 3 cards from the deck. Let X be the number of tables where at least one player has only red cards. Compute P (X = 6).

4.6 dice are Let X be the number of different numbers which occur (for example if the numbers shows are 6, 2, 3, 6, 6, 3 then X =3).

(a)Compute P (X =3).

(b)Compute EX.

### 5.Let X have denisty xe−xif x 0 and 0 otherwise.    概率导论final exam代写

(a)Compute P (X >2).

(b)Find the density of Y =X2.

6.When a crush occurs on 40 mile highway its location has densityx .

(a)Find a probability that the next crust will  occur  between  mileposts  10  and  30.

(b)Duringa certain week there was 10  Find the probability that exactly 6 occurred between mileposts 10 and 30.

(c)Let N be the first crush which appear before milepost 10. Compute EN and VN.

(d)Where on highway should a service station be located to minimize the expecteddistance to the next crush?

7.Let  (X, Y )  have  density  equal  to  cxyif x 0, y 0 and  x y 1 and equal to 0 otherwise.      概率导论final exam代写

(a)Compute E(XY).

(b)Compute P (X > 2Y).

(c)Compute the marginal distribution of X.

8.Let X and Y be independent, X have uniform distribution on (0, 1) and Y have uniform distribution on (0,2).

(a)Find the density of Z = X +Y.

(b)Find P (X > Y).

### 9.Let (X, Y ) be a random variables with EX = EY = 1, V (X) = 5, V (Y ) =10,Cov(X, Y ) = 2.

(a)Compute Cov(X +Y, X Y ).

(b)0Compute V (2X + 3Y).

(c)Compute E(2X + 3Y)2.

(d)Useone sided Chebyshev inequality to estimate P (2X + 3Y > 30).

10.Let X and Y denote tax revenue from individuals and busynesses Suppose that(X, Y is a normal vector and EX = 15, EY = 20, V (X) = 1, V Y = 9, Cov(X, Y )=2.

(a)Let T = X + Y . Find the distribution of T.

(b)Let D = Y X. Find P (D <4).

(c)Compute P (X  < 15|Y  =21).

(d)Compute P (X < 15|T  =33).

11.Let N be the number of times Jane stops at traffic lights during her drive to and from work Suppose that P (N = 1) = 0.2, P (N = 2) = 0.5, P (N = 3) = 0.3. Suppose moreover that the time of each stop is uniformly distributed on the interval from 0 to 2 minutes and that those times are independent. Let T be the total number Jane waits at the traffic light during her work day.    概率导论final exam代写

(a)Compute E(T N = 2) and V (T N =2).

(b)Compute E(T ) and V (T).

(c)Let M be the total number of times Jane is stopped at the traffic light on Monday and Compute P (M =4).

(d)Let Z be the total time Janes waits at traffic lights during a particular workweek (Monday-Friday). Compute EZ.

(e)Let D be the total number of days during a workweek Jane has exactly three Compute P (D =2).

(f) Call a week  unlucky if Jane has three  traffic light stop per  day for at least  three days.

Let L be the number of the first unlucky week. Compute EL.

### 12.Let X1, X2. . . X240be independent identically distributed random variables such thatXj has density 3x2if x [0, 1] and 0 otherwise. Let S = X1 + X2 + + X240.    概率导论final exam代写

(a)Compute ES and VS.

(b)Find approximately P (S >183).

13.Misprints in a 400 page book form a Poisson process with intensity 3misprints/page.

(a)Let X be the total number of misprints in a Compute EX and VX.

(b)Compute approximatelyP (X 1150).

Let N be the first page without misprints. Compute P (N =25).

(c)Compute a probability that in a 30 page chapter there are exactly 2 misprint free pages.

The book is read by two Each misprint is captured  by each  proofreader  with probability 0.8 independently of the other proofreader. Let Y be the total number of misprint left in the book after the proofreading. Compute EY and V Y.