Probability theory and mathematical statistics

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Section I. The classical definition of probability
From the deck of 36 cards at random, a 3 card. Find the probability that among them will be one ace.
Section II. Theorems of addition and multiplication of probabilities
At the construction site of the crane 3. The probability of failure-free operation of the first crane is 0.7, the second - 0.8, the third - 0.9. Find the probability that at least one running tap.
Section III. Conditional probability
Each of the following words made up of letters, blocks, scattered into separate blocks, which are then stacked in a box. Out of the box the letters randomly removed one by one. Find the probability of getting at that removing the old word: Marble.

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Section IV. The total probability. Bayes' formula
There are two batches of parts, and it is known that no defective first and second defective 1/4. Detail taken from the randomly selected part, was nebrakovannoy. What is the probability that it is taken from the second installment?
Section V. The Bernoulli scheme. Limit theorems in the Bernoulli scheme
In the television studio cameras 3. For each camera a possibility that it is on at the moment is 0.6. Find the probability that at the moment includes two cameras.
Section VI. Distribution function and probability density functions. Numerical characteristics of random variables
The random variable X is given by the distribution function
F (x) =
Find the distribution density f (x), the expectation M (X) and the variance of D (X).
Section VII. The sample, its numerical characteristics
For the following statistical distributions of samples required:
1) Find the empirical distribution function and construct its graph.
2) Construct a frequency polygon.
3) Calculate the sample mean.
4) Evaluate selective dispersion and corrected.
xi April 1 August 10
ni 5 3 2 1
Section VIII. The linear correlation
According to the data given below, calculate the coefficient of correlation, find sample regression equation of a straight line Y on X, build correlation field and apply it to the regression line Y X.
X 5 First 9 10 12
Y 3 6 4 7
Section IX. Statistical verification of statistical hypotheses
Empirical distribution of a discrete random variable X. required, using the criterion test for a significance level of = 0.05 hypothesis about the distribution of the population by the Poisson law.
The random variable X - the number of failed machines in the shop for one shift. Number of observation n = 200:
xi 0 1 2 3 4 5 6 7 8
ni 41 62 45 22 16 8 4 2 0
Section X. Confidence intervals for the parameters of the normal distribution
Find the confidence interval for the estimate of the expectation as the normal distribution with the reliability of 0.95, knowing the sample mean = 75.17, sample size n = 36 and standard deviation σ = 6.


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