Download Self-training 5

Ministry of Health Care of Ukraine
O.O.Bohomolets National Medical University
EXERCISE BOOK 5
for independent work of students
Study discipline «Medical and biological Physics»
Specialization «Medical occupation», «Stomatology»
Department of Medical and biological Physics
Authors: A.V.Chalyi, N.V.Stuchynska, I.F.Margolych
Approved by methodical counsel of the Department of Medical and Biological
Physics on 31.08.2015., proceeding № 1
Considered and approved by CMC of physical and chemical disciplines on
1.09.2015., proceeding № 1
Topic 5. Elements of mathematical statistics
Education aims:
- to master notion of discrete and continuous random variables;
- to familiarize with the distributions of discrete and continuous random
variables;
- to give the notion of main numerical characteristics of distribution of
random variables;
- to show the applications of main numerical characteristics of distribution of
random variables.
Abilities and skills:
- to know determination of discrete and continuous random variables;
- to be able to find main numerical characteristics of distribution of random
variables: mathematical expectation, variance, standard deviation;
- to know the formula of normal distribution;
- correlation dependence between random quantities;
- coefficient of correlation;
- general set and sample;
- point and interval evaluation of parameters of general set.
Literature:
1.Medical and Biological Physics: Textbook for the students of higher medical
institutions of the IV accreditation level/ Chalyi A.V., Tsekhmister Ya.V., Agapov
B.T.[et al.]. Vinnitsja, Nova Knyha, 2010. - 480 pp.
2
2. Chalyi O.V., Stuchynska N.V., Melenevska A.V. Higher mathematics. Train aid
for the students of higher medical and pharmaceutical institutes. – K., Теhnica.2001.
3. Hoffmann L.D., Bradley G.L. Calculus for business, economics, and the
social and life sciences New York, St. Louis, San Francisco, McGraw-Hill Inc. –
5th. ed. , 1992.-778 pp.
4. Calculus. A Committee of Lebanese authors. Beirut, Librairie du Liban.3th.ed., 1997.-291 pp.
Control questions for theoretical study:
1. What is discrete random variable?
2. What is continuous random variable?
3. What is called distribution of random variable?
4. What are main numerical characteristics of distribution of random
variables?
5. Give the formulae of main numerical characteristics of distribution of
random variables.
6 Write the formula of density distribution function of normal distribution.
7. Write the formula for calculation of correlation coefficient.
8. General set and sample.
9. Point and interval evaluation of parameters of general set.
Tasks for the independent work:
Task 1.
Calculation of mathematical expectation, variance, and standard
deviation.
1. The distribution law of some random variable is given by the table:
X
1
3
4
6
7
P
0.2
0.1
0.2
0.3
0.2
Find mathematical expectation, variance, and standard deviation.
2. The distribution law of some random variable is given by the table:
3
X
2
3
4
7
9
P
0.1
0.2
0.3
0.2
0.2
Find mathematical expectation, variance, and standard deviation.
3. The distribution law of some random variable is given by the table:
X
-1
0
1
2
3
P
0.2
0.2
0.3
0.1
0.2
Find mathematical expectation, variance, and standard deviation.
4. The distribution law of some random variable is given by the table:
X
-3
-1
0
2
3
4
P
0.1
0.2
0.3
0.2
0.1
0.1
Find mathematical expectation, variance, and standard deviation.
5. The distribution law of some random variable is given by the table:
X
2
3
5
7
9
P
0.2
0.1
0.4
0.2
0.1
Find mathematical expectation, variance, and standard deviation.
6. The distribution law of some random variable is given by the table:
X
-5
-2
1
4
8
P
0.1
0.2
0.4
0.2
0.1
Find mathematical expectation, variance, and standard deviation.
7. The distribution law of some random variable is given by the table:
X
2
4
7
10
12
P
0.2
0.4
0.2
0.1
0.1
Find mathematical expectation, variance, and standard deviation.
8. The distribution law of some random variable is given by the table:
X
-10
-6
0
5
11
P
0.2
0.3
0.3
0.1
0.1
Find mathematical expectation, variance, and standard deviation.
9. The distribution law of some random variable is given by the table:
4
X
1
5
8
12
15
P
0.2
0.2
0.3
0.2
0.1
Find mathematical expectation, variance, and standard deviation.
10. The distribution law of some random variable is given by the table:
X
2
7
10
13
15
P
0.1
0.3
0.3
0.2
0.1
Find mathematical expectation, variance, and standard deviation.
Task 2. Ascertain existence of correlation dependence between random quantities.
1. Find existence of correlation dependence between random quantities Х and
Y:
Х
Y
5
0,1
10
0,6
7
0,2
6
0,5
2. Find existence of correlation dependence between random quantities Х and
Y:
Х
Y
3,3
1,7
1,6 3,5
1,2 3,0
2,9
2,7
3. Find existence of correlation dependence between random quantities Х and Y:
Х
Y
2
0,1
8
0,3
12
0,2
15
0,5
18
0,8
4. Find existence of correlation dependence between random quantities Х and Y:
Х
Y
10
0,5
20
0,7
26
0,6
31
0,8
40
1,1
45
1,4
Task 3. Find confidential intervals for mathematical expectation with given
confidential probability.
1. The distribution law of some random variable is given by the table:
X
P
1
0.2
3
0.1
4
0.2
6
0.3
7
0.2
5
Find confidential intervals for mathematical expectation with confidential
probability 0,95.
2. The distribution law of some random variable is given by the table:
X
2
3
4
7
9
P
0.1
0.2
0.3
0.2
0.2
Find confidential intervals for mathematical expectation with confidential
probability 0,99.
3. The distribution law of some random variable is given by the table:
X
-1
0
1
2
3
P
0.2
0.2
0.3
0.1
0.2
Find confidential intervals for mathematical expectation with confidential
probability 0,95.
4. The distribution law of some random variable is given by the table:
X
-3
-1
0
2
3
4
P
0.1
0.2
0.3
0.2
0.1
0.1
Find confidential intervals for mathematical expectation with confidential
probability 0,95.
5. The distribution law of some random variable is given by the table:
X
2
3
5
7
9
P
0.2
0.1
0.4
0.2
0.1
Find confidential intervals for mathematical expectation with confidential
probability 0,99.
6. The distribution law of some random variable is given by the table:
X
-5
-2
1
4
8
P
0.1
0.2
0.4
0.2
0.1
Find confidential intervals for mathematical expectation with confidential
probability 0,99.
7. The distribution law of some random variable is given by the table:
X
2
4
7
10
12
P
0.2
0.4
0.2
0.1
0.1
Find confidential intervals for mathematical expectation with confidential
probability 0,95.
8. The distribution law of some random variable is given by the table:
X
-10
-6
0
5
11
P
0.2
0.3
0.3
0.1
0.1
Find confidential intervals for mathematical expectation with confidential
probability 0,999.
9. The distribution law of some random variable is given by the table:
6
X
1
5
8
12
15
P
0.2
0.2
0.3
0.2
0.1
Find confidential intervals for mathematical expectation with confidential
probability 0,95.
10. The distribution law of some random variable is given by the table:
X
2
7
10
13
15
P
0.1
0.3
0.3
0.2
0.1
Find confidential intervals for mathematical expectation with confidential
probability 0,99.
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