3. Let the correlation coefficient of random variables X and Y be 5.0, () () 0, EXEY2.
2
() () 2EXEY, then
2
EXY
.
4. Let the random variable x obey the parameter? Poisson distribution of sum 3
1}0{XP, then.
5. Set the whole?
2
,
~?
NX， 1
2(，)X
X is a sample extracted from X, and the sample size is 2, so the joint probability density function of 12 (,) XX is 12, gxx? _________________________.
6. let the whole x obey the parameters? Exponential distribution of () e? ，nXXX，，，2
1? Is a simple random sample in the population x, so () DX? .
7.Let ] 1，[~aUX，nxx， 1？ Is a sample extracted from the population x, and the moment of a is estimated to be. 8. if X~()tn, then x
2
~ .
Second, multiple-choice questions (3 points for each question, ***24 points)
1, right? A ball, randomly placed in N boxes (N), is designated? The probability that there is a ball in each box is. (1)
n
!
(B)？
n
chrome
n
!
(3)
n
n？
!
(D) n
n
nC？
!
2, set 8.0)| (,7.0) (,8.0) (? BApBpAp, then the following conclusions are correct: (A) A and B are independent of each other; (b) Events A and B are mutually exclusive; (C)AB？ (D) )()()(BpApBAp？ 3. Let the probability density of the random variable X be |
|)(xce
Xf, then c = (A)-
2
1 (B)0 (C)
2
1 (D)
4. Let x obey parameter 9.
1?
Exponential distribution) (xF is its distribution function, then? }93{XP ()) (1) 9
3() 1(FF？ ; )(B
) 1 1
(
nine
13
e
e
; )
(Ce
e
1 1
three
; )
(D.9)
/30
xedx
5. Let x and y be two random variables, 7.
300?
YXP，，7
400?
YPXP, then 0maxYXP,
A
seven
5; B
Forty nine
16; C
seven
3; D
Forty nine
40
6. Suppose that the random variables X and Y are independent and identically distributed, and write down YXU and YXV, then there must be an independent B correlation coefficient between U and V that is zero, and the C correlation D correlation coefficient is not zero. 7. Let NXX, 1? Is a sample in the population x, and () EX, then the following is? The unbiased estimate of is ()
)
(A？ 1
1
1
niiXn
; )
(B？ n
iiXn 1
1
1
; )
(Cn
iiXn
2
1
; )
(D 1
1
1
1
niiXn
8、 162 1，，，XXX？ From the population ~(0 1XN,)
A simple random sample, assuming: 2
2
18ZXX 2
2
9 16YXX, then y
(A) 1，0(N)(B) 16(t)(C) 16(2
(D)8，8(F)
(3) 1, (6 points) If AFP is used to diagnose liver disease, the probability that people with known liver disease are diagnosed with liver disease is 0.95, and that of people without liver disease is 0.02. Suppose the incidence of liver disease in the population is 0.0004, and now there is a person who has been diagnosed with liver disease. Find the probability that this person is indeed a patient with liver disease.
2.(6 points) Let the random variable 12, and the probability distribution of XX is
10 1
1 1 14
2
four
I
empirical value
1，2i？ And it satisfies 12(0) 1PXX, and the joint distribution table of 12 and the correlation coefficient of XX are 12 (,) RXX.
3.( 14 points) Let the random variables X and Y obey uniform distribution in the region D, where D is surrounded by 1, 0, xxyxy, and find the joint density function of (1)X and Y; (2) The edge distribution of X and Y, and discuss whether X and Y are independent; (3) Expected (value of XYE).
4.(6 points) A bus will carry 25 passengers to 9 stations, and each passenger is waiting to get off at each station, and whether to get off or not is independent of each other. Traffic only stops at stations where people get off. Mathematical expectation of finding the number of stops x of traffic vehicles.
5.(8 points) The average pulse rate of normal people is 72 times per minute, and the pulse rate of 10 patients with elixir poisoning has been measured.
The average number of calculations is 67.4, and the mean square error is 5.929. It is known that human pulse frequency obeys normal distribution. Is there any significant difference between poisoned patients and normal people?
(Possible number: 0.025(9)2.262t? ，0.05(9) 1.833t？ ，0.025( 10)2.23t？ ，0.05( 10) 1.8 12t？ )
6.( 12 points) Let the global x density function be 2.
2，0()0，x
xfx
other
， 12，，，nxxx？ For samples from the population,
Beg? Moment estimation and maximum likelihood estimation.
One,
1、
five
2; 2、
12
; 3、6; 4、3ln5、
22
122
()()
22
12xxe
; 6、
2
1?
n；
7、
12
1
n
two
x
n
Or 2 1x? ; 8 、( 1，)Fn
Second, 1, a; ; 2、A； 3、C； 4、C； 5、A； 6、B； 7、D； 8, D 3, 1, (6 points) solution: let A={ liver disease patient}, B={ diagnosis of liver disease}, by Bayesian formula,
)
|()()|()()
|()()|(abpapabpapabpapapbap
3 points
.0 187.002
.0)0004.0 1(95.00004.095
.00004.0?
3 points
Construction engineering application certification! Double the value of wealth, giving priority to exclusive display of peer exchanges.
2.(6 points) Solution: The joint distribution of 12 (,) XX is
X2
x 1– 1
1
– 1
1
four
1
four
0 1
four
14
12 1 0
14
14
14
12
14
12 120，020,0,0EXEXEXX？ , so 12cov (,) 0XX? So 12 (,) 0RXX? .2 points
3.( 14 points) Solution: (1)1|)] [1]
2
1
XdxxxS, so?
0 1
(xf other Dyx? ), (4 points xdyyxfxfx
xX2 1)，()(？
?
02)(xxfX other 10x dxyxxfyfy?
() (When 0 1? y，
ydxyfyY？ 1 1)( 1
When 10y, ydxyfy
Y 1 1)( 1
therefore
0 1 1)(yyyfY other 100)
1? Yy is not independent because), () (yxfyfxfYX). (3)00)，()( 1
1
Advanced (short for deluxe)
xydy
dxdyyxxyf
dxXYEx
x
2 points
4.(6 points) Solution: Let 0 1.
No passengers got off at nine stations.
Bus * * * passengers get off at the first stop, and bus * * * is at the second (1, 2, 9i) and 9.
1
iiXX
}0{0} 1{ 1iii
XPXPEX
25
eight
1()9 9
25 1
eight
() () 9 [1()] 9 iiex ex point 2
5, (8 points) solution: from the meaning of the question,), (~2
NX H0:720h 1:720) 1(~/
ntn
Where is SXT? 929.5,4.67, 10? SXn is replaced by 2622.2) 9 (453.38+00
/929.5724.67025.0?
Tt So, reject H0 and think there is.
Significant difference. 2 points 6, (12 points) solution 2
2
223
x
Edx？
By 2 o'clock
three
X, so
32
x
Likelihood function 1 122
(，)n
nnn
Lxxxx
1
lnln22lnlnn
I
iLnnx
ln2dLn
d？
So () l? Monotone decreasing
^
1，maxLnxx