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Special topics on mathematical statistics and probability in senior high school
Review questions of probability and statistics

1, with three boxes numbered 1, 2,3.1. Box 0 contains 1 red balls and 4 white balls, box 2 contains 2 red balls and box 3 contains 3 red balls. Someone takes any box from three boxes, finds a ball from it, and finds the probability of getting a red ball.

2. A, B and C shoot at the plane at the same time, and the probability of three people hitting is 0.4, 0.5 and 0.7 respectively. The probability of the plane being shot down by one person is 0.2, and the probability of being shot down by two people is 0.6. If all three people hit it, the plane will definitely be shot down. Find the probability of the plane being shot down.

3. There are three boxes numbered 1, 2,3, 1. Box 0 contains 1 red balls and 4 white balls, box 2 contains 2 red balls and 3 white balls, and box 3 contains 3 red balls. Someone took any one of the three boxes, found a ball and found it was a red ball. The ball is 65438.

4. Shops sell glasses by the box, and each box contains 20 glasses, of which each box contains 0, 1 one, and the probability of two defective glasses is 0.8, 0. 1, 0. 1 respectively. A customer chose a box and four glasses from it. The results were all good, so he bought the box. What are the chances of defective glass in this box?

5. There are products of the same brand produced by three factories, A, B and C. It is known that the market share of the three factories is 1/4, 1/4 and 1/2 respectively, and the defective rate of the three factories is 2%, 1% and 3% respectively. Try to find brand products on the market.

6. Let the density function of x be and find the probability density of Y=2X+8.

7. Let the distribution law of random variable x be:

X -2 - 1 0 1 3

p 1/5 1/6 1/5 1/ 15 1 1/30

Find the distribution law of Y = X 2

8,

9. Let the probability density of (x, y) be

Find the value of (1) c; (2) Density of two sides.

(3) Are X and Y independent?

10, let the probability density function of random vector (x, y) be

Try to judge whether x and y are independent of each other.

1 1, if x and y are independent of each other and obey Poisson distribution with parameters respectively, it is proved that Z=X+Y obeys parameters.

Poisson distribution.

12,

13 and find the distribution rate of 2X+3.

14, let X 1, X2, …Xn be a sample in the population X~B( 1, p), and find the maximum likelihood estimator of the parameter p. 。

15, let the population x obey the uniform distribution on [a, b], a and b are unknown,. x 1, x2...xn are samples from x, and try to find the moment estimators of a and b. 。

16, let the length x of a part obey the normal distribution N(μ, 0.42). Now, 20 pieces are selected, with an average length of 32.3 mm, and the confidence interval of their length is found with 95% confidence.

17 has a lot of sweets. Now we randomly select 16 bags and weigh them as follows (grams):

506 508 499 503 504 5 10 497 5 12

5 14 505 493 496 506 502 509 496

Assuming that the weight of bagged candy approximately obeys normal distribution, try to find the confidence interval of the overall average confidence level of 0.95.

The radiation of 18 microwave oven when the oven door is closed is an important quality index. This quality index of a factory obeys normal distribution, and the average value meets the requirements for a long time, not exceeding 0. 12. In order to check the quality of the products in the near future, 25 sets were randomly selected, and the average radiation amount when the oven door was closed was obtained. Does the radiation increase when the oven door is closed horizontally?

19, a sugar factory uses automatic baler to pack, and the standard weight of each bag is 100 kg. After starting work every day, it is necessary to check whether the baler works normally. After starting work on a certain day, the weight of nine bags is measured as follows:

99.3,98.7, 100.5, 10 1.2,98.3,99.7,99.5, 102. 1, 100.5

Assume that the weight of each package obeys a normal distribution. When the significance level is 0, does the baler work normally?

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