Test paper number: (b) Volume

Course Name: Probability Theory and Mathematical Statistics Applicable Class: Undergraduate

College: Department: Examination Date: June 65438+ 10/0, 2006.

Major: Class: Student ID: Name:

The title number is one, two, three, four, five, six, seven, eight, nine, and the total score is signed.

Topic 100

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Fill in the blanks (3 points for each blank, *** 15 points)

1 If P (a) = P (b) = P (c) =, P (AB) = 0, P (AC) = P (BC) =, then A, B and C will not happen.

The probability is.

2 Let the mathematical expectation E(X)= and variance D(X)=2 of random variable X, then Chebyshev inequality.

We can see P{|X-|3}.

3. Let the probability density function of random variable X be f(x)=, then the mathematical expectation of X is.

There are five students in one group, so the probability that their birthdays are different is. Suppose a year is 365 days.

5. Let f (x), g (x) and h (x) all be probability density functions, and the constants A, B and C are not less than zero, so,

Af(x)+bg(x)+ch(x) is also a probability density function, so there must be a+b+c=.

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Multiple choice questions (3 points for each question, *** 15 points)

1..a and b are random events, BA, then the following formula is correct ().

(A)、P(AB)=P(A) (B)、P(B-A)=P(B)-P(A)

P(AB)=P(A) (D)、P(B)=P(B)

2. People's weight X obeys a certain distribution, e (x) = a, d (x) = b, and the average weight of 10 people is recorded as y, so there is.

(A) E(Y)=a，D(Y)= b； (B) E(Y)=a，D(Y)= 0. 1b；

(C) E(Y)=0. 1a，D(Y)= b； (D) E(Y)=0. 1a，D(Y)=0. 1b。

3. Assuming that events A and B satisfy P(B/A)= 1, then

(A)A is an inevitable event, (B)P(/A)=0, (C)A, (d) A.

4. If the random variables X and Y are independent, then ()

a、D(X-3Y)=D(X)-9D(Y) B、D(XY)=D(X)D(Y)

C, D,

5. Let the random variables X and Y be independently and identically distributed, U = X-Y and V = X+Y, then the random variables U and V are inevitable.

Not independent; Independence; (c) The correlation coefficient is not zero; The correlation coefficient is zero.

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3. Set the distribution function of two-dimensional continuous random variable (x, y).

F(X, Y)= A(B+ arctangent) (C+ arctangent)

Find (1) coefficients a, b, c.

(2) Probability density of (x, y);

(3) Edge distribution function and marginal probability density. (12)

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4. Let the probability density of random variable X be

f(x)= 1

Now, X is repeatedly observed independently for n times, and Vn is used to indicate the number of times the observation value is not more than 0. 1, and try to find the distribution law of Vn. ( 10)

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5. In artillery battle, the shooting probability at the distance of 250m, 200m and 150m is 0. 1, 0.7 and 0.2, respectively, while the probability of hitting the target is 0.05, 0. 1 and 0.2, respectively, thus the probability of the target being destroyed can be obtained. If the target is known to be destroyed, find the probability of destroying the target.

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6. Let x and y be independent random variables, and the probability densities are respectively

fX(x)=，fY(y)=，

Try to find the probability density of z = 2x+y (12 minute).

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Seven, a box of 100 products, of which 80, 10, 10 are the first, second and third types of products respectively. Now choose one at random and record (i= 1, 2, 3). Try asking:

Joint distribution law of (1) random variables; (2) Joint distribution function of random variables; (3) Correlation coefficient of random variables. (12)

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8. Let the probability density function of random variable X be

Find: (1) constant (10 point)

nine

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Let events A, B and C be independent, and prove that AB, AB and A-B are independent. (4 points)