Control 3 and 4 on probability theory, option 6

Examination №3

1. The probabilities that each of the three tellers busy with customers, are, respectively, 0.7; 0.8; 0.9.

Find the probability that at the moment are busy shoppers:

a) all cashiers;

b) only one cashier;

c) at least one cashier?


2. On the correspondence department of the university 80% of all students working in their specialty.

What is the probability that the five randomly selected students in the specialty work:

a) two students;

b) at least one student?


3. In the e-mail received 8,000 letters. The probability that a randomly taken envelope no index is 0.0005.

Find the probability that the zip code is missing:

a) in three envelopes;

b) not less than three envelopes.


4. commercial agent has five addresses of potential buyers, which he refers to the proposal to buy the company implemented its goods. The probability of agreement of potential buyers, respectively, as estimated 0.5; 0.4; 0.4; 0.3; 0.25. Agent addresses them in the order listed until until someone agrees to buy goods.

Create the distribution law of the random variable - the number of buyers, which will have to consult dealer. Find the mean and variance of the random variable.


5. The probability density normally distributed random variable X has the form:

Search:

a) the mean and standard deviation of the random variable X;

b) the probability;

c) the likelihood that the deviation of the random variable X from its mathematical expectation exceeds 2.5 (in absolute value).


Examination №4

1. Are selected data on the distribution of the size of the contribution of depositors in the Savings Bank of the city

The size of the contribution, thous. Up to 40 40-60 60-80 80-100 More than 100 Subtotal

The number of contributions 32 56 92 120 100 400

Search:

a) the probability that the average size of deposit Savings differs from the average size of the contribution to the sample of not more than five thousand. (In absolute value);

b) border, which concluded with a probability of 0.95 share of deposits, the amount of which less than 60 thousand rubles .;

c) the volume of re-sampling, in which the same limits for the share of deposits (see. PB) can be guaranteed with a probability of 0.9876; to give an answer to the same question, if no preliminary data is not considered share.


2. According to the task 1 using - Pearson, at the level of significance to test the hypothesis that the random variable X - the size of contributions to the Savings Bank - normal distribution.

Build on one drawing a histogram of the empirical distribution and the corresponding normal curve.


3. The distribution of 110 companies the value of fixed assets X (mln.) And the value of output Y (million rubles.) Presented in the table.

y

x 15-25 25-35 35-45 45-55 55-65 65-75 Total

5-15 17 April 21

15-25 3 18 3 24

25-35 2 15 5 22

35-45 3 13 7 23

45-55 6 14 20

Total 20 24 21 18 13 14 110

You need:

1. Calculate the average group xi and yj, build empirical regression line.

2. Assuming that between the variables X and Y, there is a linear correlation:

a) Find the equation of the regression lines, build their schedules on a drawing with empirical regression lines and give an economic interpretation of equations;

b) calculate the correlation coefficient; at the level of significance to assess its value and draw a conclusion about the direction of distress and relationships between variables X, and Y;

c) Use the appropriate regression equation to determine the average value of output, if the value of fixed assets is 45 million.