Теория вероятностей on Windows Pc

CAUTION Possible errors!

Questions:

1. The concept of the space of elementary events. Examples. Random events.
2. The classical definition of probability. Properties probabilities of events.
3. The axiomatic definition of probability. Prove the corollary of the definition.
4. Derive the formula of total probability and Bayes' formula.
5. Derive the formula of Bernoulli and consequences of it. (For the probability of success of k to m 0 and the probability of success.)
6. The conditional probability. Multiplication theorem. Independent events.
7. Prove the criterion of independence of two random events.
8. Formulate a definition of a discrete random variable, to substantiate its view of the distribution function.
9. The distribution function of NE and its properties.
10. The probability density function and its properties.
11. Define the binomial distribution and a Poisson distribution. Establish a link between them. (Binomial tends to Poisson when n → ∞, np → λ.)
12. Random vectors. The distribution function of the random vector and its properties.
13. The density of the multivariate random vector and its properties.
14. Functional transformations ST. Determination of the distribution of the functions of the known distribution law argument. Consider a special case: X2 = φ (X1), where φ monotonic function.
15. Conclusion of the formula for the composition of a distribution.
16. Numerical characteristics of random vector.
17. The correlation coefficient and its properties.
18. Conditional distributions. Derive an expression for the conditional density f (Y | X).
19. Expectation and its properties.
20. Formulate the LLN. To prove the theorem of Chebyshev.
21. Prove Theorem Bernoulli (as a consequence of Chebyshev's theorem).
22. Formulate the central limit theorem and deduce (as a consequence) Moivre-Laplace theorem.
23. Print the Chebyshev inequality and formulate the law of large numbers in the form of Chebyshev.
24. Custom and empirical distribution functions and their properties.
25. The empirical distribution density and its properties.
26. Estimation of the parameters of the distribution. Point estimates. Requirements for the point estimate.
27. Show that X is an unbiased, consistent and efficient estimates in the class of linear estimators.
28. Prove that 1 / n * sum (X_i - xcp) 2 is a biased estimator of the variance.
29. The method of maximum likelihood.
30. Find the maximum likelihood estimation of the parameters of the normal distribution.
31. Find the maximum likelihood estimate of the parameter of the exponential distribution.
32. Find the maximum likelihood estimate of the parameter of the binomial distribution.
33. Determination of the confidence interval (CI). Its probabilistic sense.
34. Build a CI for the mat. expectations normally distributed with known SW rms
35. Build a CI for the mat. expectations normally distributed with unknown NE rms
36. Building a CI for the mat. waiting for the unknown variance.
37. Derive an expression for the CI for the dispersion and rms normally distributed ST.
38. An optimal criterion for mat. expectations normally distributed general population with known variance for the case of two simple hypotheses.
39. Testing statistical hypotheses. Errors of the 1st and 2nd kind. The notion of testing hypotheses. Critical region, the level of significance.
40. Rule Neyman-Pearson constructing the best critical region. Give an example.
41. Criteria for testing hypotheses about the equality of the two middle NHS under certain rms
42. Testing hypotheses about the variance of normal general population (NGS) and the equality of two variances NHS.
43. The notion of goodness of fit. Pearson's chi-squared test and its application.
44. The task of smoothing the experimental dependence. The method of least squares parameter estimation of a linear model.
Combination and placement
Statistics and critical sets

In practice, when the conditional probabilities are dividing by the square of sigma (sigma requires a) - error.