Probability statistics: Fifth chapter large number law and central limit theorem _ law of large numbers

Fifth chapter The law of large numbers and the central limit theorem

Content Summary:

The law of large numbers

Theorem one (the special case of Chibi's law of large numbers) sets random variables to be independent of each other, with the same mean and variance:

, for any given positive number, there are

Set a random variable is a sequence of random variables, is a constant. For any given positive number, there are

The said sequence converges to the probability and is recorded as

The nature of a sequence of convergent probabilities:

Set the function in a continuous point, then

Theorem one can also be described as:

Set random variables to be independent of each other, with the same mean value and variance:

Then the sequence converges according to probability.

Theorem two (Bernoulli's law of large numbers) is set to be the number of events in an independent repetition test, the probability that the event occurs in each trial, and for any given positive number, there

or recorded as

.

Theorem Three (Sinchin law of large numbers) to set random variables independent of each other, with the same distribution, then for any given positive number, there are

.

Second, central limit theorem

Theorem four (independent of the central limit theorem of distribution) to set random variables independent of each other, with the same distribution, remember

For any real number, there are

Theorem Four shows: The distribution function of the normalized variable of the homogeneous random variable with the variance of the independent same distribution, when sufficiently large, has

Or

Theorem Five (Lyapunov's theorem) sets random variables to be independent of each other, with mathematical expectation and variance:

Remember

If there is a positive number, so

The normalized variable of the random variable:

For any real number, the distribution function satisfies

Theorem VI (Mover-Laplace theorem) sets random variables to obey two-item distributions. For any real number, there are

Analysis of typical examples

Example 1, to set the random variable independent, and uniform distribution on the same subject, proved

ID: For any integer, if,

Was

The

So

Example 2, to set the random variable independent of each other, and the uniform distribution on the same subject, is a continuous function. Trial Proof

Evidence: Because of the same distribution of independence, the

are independent and have the same distribution, and

By Sinchin Law of large numbers,

That

Example 3, the statistics of an insurance company for many years indicate that the number of claims stolen by the claimant households is accounted for in the case of the 100 claimants who have been randomly sampled and claimed by the insurance company. Claim the stolen claimant households not less than 14 households, but not more than 30 of the approximate probability.

Solution: By the asked, according to the Mover-Laplace theorem,

Self-Test questions

1, (5 points) to set the random variable to follow the Poisson distribution parameters, then

。

2, (15) on the scale of repeated weighing a heavy item, assuming that the results of each weighing is independent of each other, and the same obeys the mean, the variance of the distribution, representing the weight of the results of the arithmetic average value, the minimum positive integer.

3, (10) There are 300 paging stations in a certain city, the number of calls received by each paging station in each minute obeys the Poisson distribution of the parameters, and the probability of the total number of calls within a minute of the city being less than 70 times is tried.

4, (10) throw a dice, in order to have at least 95% grasp the point 6 upward frequency and probability of the difference between the absolute value within the range, asked how many times to throw.

5, the computer in the Addends, to each addends (take the nearest integer), set all the integer error independent of each other, and they all obey the uniform distribution, if the sum of 1500 numbers, ask the total number of errors in the absolute value of more than 15 of the probability is.

(d) The reference answer to the self test question

(1) (2) 16 (3) (4) 5336 (5)

From:http://lxy.cumtb.edu.cn/gailvtongjidaoxue/chap5.htm