概率论与数理统计英文题目

Test
1. Consider the set {}9,8,7,6,5,4,3,2,1=Ω with subsets {}{}{}{}8,7,4,3,2,1,8,6,4,2,
9,7,5,3,1====D C B A
Find the following sets:
(a) D A and B A (b) c D C )( and )(D B A
2. ShowDe Morgans Law: (a) c
c c F E F E =)(,(b)c c c F E F E =)(
3.(a)Let P(A)=,P(B)=,P( A∩B)=. Find P(A∪B) and P(A|B)
(b)Consider two fair dice A and B. Die A is six-sided and is numbered 1 through to 6 whilst die B is four-sided and is numbered 1 through 4. Both dice are rolled. Find the probability of two dice show the same score. 4.(a).Let X be a random variable. ),(~2σμN X . Show that
)1,0(~N X Y σ
μ
-=
(b)The pmf of a random variable X which has a Poisson distribution with parameter 2. Find P(X=3).
5. Suppose (X,Y) be bivariate normal distribution.
);,,,(~),(222211ρσμσμN Y X . Find the correlation coefficient of X
and Y.
6. (a)Let X Uniformly distributed on (-1,1), . X~U(-1,1). Find the Expectation and Variance of X.
(b)Let X and Y be continuous random variables with joint pdf
.),(),(2,R y x y x f Y X ∈ Let Z=X+Y, Find )(z f Y X +. In particular, if X
and Y are independent, Find f X *f Y..
7.(a)The random variable X has pmf is: X 0 1 2 3 P X (x)
Find P(1≤X<3) and F X .
(b)Let X be a continuous random variable with pdf
⎩⎨
⎧≥=-otherwise x e x f x 0
2)(2 Find P(X=3), P(X>1) and )12(>>X X P . 8. X is continuous . with pdf
2
210
2()10
2
x
x
e x
f x e x -⎧≤⎪⎪
=⎨⎪>⎪⎩
Let X Y =, Find E(Y)
X be a continuous random variable with probability density function
⎩⎨
⎧>=-otherwise x xe x f x 0
)( (a) Find P(1<X<2) (b) Find: (1) E(X); ,(2) Given that E(X 2)=6. Find Var(X)
10．If the joint probability density function of two random variables is given by ⎩⎨
⎧>>=--otherwise y x e x f y x 0
0,06)(32
(a) Find the marginal pdf of X and Y.
(b)Determine whether the two random variables are dependent or independent.
11．Suppose the random variable X has the density function
⎪⎩
⎪
⎨⎧>+=otherwise x x x f 00)4(32
)(3
Find the probability density function of the random variable Y=X+4. 12. Take out a number from 1-200 ( 200 natural numbers).
Find (a) the probability that this number can be drived by 6;
(b) the probability that this number can be drived by 6 as well as 8.
13. Three children are selected at random from a group of five boys and three girls.
(a)What is the probability that all three are boys?
(b)What is the probability at least two girls are selected?
14. (a)Let ).(,)(),()(B findP p A P B A P AB P ＝= (b)Let 5
1)(=A P ，32)|(=
B A P ，5
3
)|(=A B P .Find )(B P . 15.(a) Suppose A, B are two events, and P(A)=1/4, P(B)=1/2, P(AB)=1/9, then evaluate ()P AB
(b)Use Bayes’ Theorem to show that if P(A),P(B)>0 and )|()(B A p A p < then )()(A B P B P <
16.(a)Let X has a Binomial distribution with parameters n and p, . X ~ b (n, p).Find )2(=X P
(b) Show that )()()(),(Y E X E XY E Y X Cov -=
17. Suppose the distribution function of a random variable X is
20,0(),011,1x F x x x x <⎧⎪
=≤≤⎨⎪>⎩
，
Find (a) the probability that X gets value within ，;
(b) the density function of X
18. The operational lifetime X, in years, of a battery powered watch has probability density function
⎩⎨
⎧≤≤-=otherwise x x cx x f 063)6()( (a) Find the value of c.
(b) Find the cumulative distribution function of X.
(c) Find the probability that the watch has an operational lifetime in excess of 4 years.
Find a and (3.2)X F
(b) A continuous random variable X having the probability density function 212()3
0x x f x elsewhere
⎧-<<⎪=⎨⎪⎩.Find ()f x dx +∞
-∞⎰ and P(0<X<1).
20.(a)Let )(3
1
)(),()(B P A P A B P A B P ===.FindP(AB) (b)
Suppose
X
and
Y
are independent random variables
),3(~),,2(~p B Y p B X ,and .9
5
)1(=≥Y P Find ).1(≥Y P
21. Suppose the density function of (X,Y) is
()()6,02,24
,0
,k x y x y f x y --<<<<⎧⎪=⎨
⎪⎩else (a) Determine the constant k. (b) Find the probability P{X<1,Y<3}.
22. Let X denote the number of times a certain numerical control machine
will malfunction: 1,2,or 3 times on any given day. Let Y denote the number of times a technician is called on an emergency call. Their joint probability distribution is given as Table Table
X
Y 1 2 3 1 2
3 0
(a) Evaluate the marginal distribution of X and Y
(b) Determine whether the two random variables of X and Y are dependent or independent.
23.(a) If 2~(20.4)X N -，
, then find 2(3)E X + (b) Let (X,Y) be 2-dimensional random variables,and
(,)(1,0)(1,1)(2,0)(2,1)
0.40.2X Y P a b
If ()0.8E XY =，find Cov(,)X Y .
the probability distribution of i X is as follows,and 1)0(21==X X P . Find the correlation coefficient of X and Y.
25. Let X has a be uniform distribution on the interval (0,1),and 142+-=X X Y Find (a))(y f Y (b) E(Y)
26. The random variable X, for fixed 0<p<1 and n ≥1,has probability mass function (pmf)
⎩⎨
⎧==otherwise
n k cp x p k
x 0
,...,2,1)( (a)Find c in terms of p and n.
(b)Find the cumulative distribution function of X.
27. If ~[0,1]X U ，~[0,1]Y U ,X and Y are independent. Find ||E X Y -.
28. The length of time, in minutes, for an airplane to obtain clearance for take off at a certain airport is a random variable Y=3X-2, where X has the density function
⎪⎩⎪⎨⎧>=-.,00,4
1)(4
/elsewhere x e
x f x
Find the mean and variance of the random variable Y.
29. Suppose the probability density function of random variable X is
1
(1)19()32
0..x x f x o w ⎧-<<⎪=⎨⎪⎩,
and 1
(1)2
Y X =-.
Find (a) the probability density function of Y,)(y f Y .
(b) expectation E(Y) and variance Var(Y).
30. Suppose X and Y are two random variables. The joint probability density function is
⎩⎨
⎧≤≤≤≤=otherwise
y x xy
y x f 0
1
0,104),(
Find (a) the marginal density of X.
(b) expectation E(Y) and variance Var(Y).
(c)Determine whether the two random variables are dependent or independent ？。