3. seminar
Problem 1
The given values 10,12,8,16,9 are considered to be the realization of the random sample
from the distribution with the expected value , the variance 2
and the distribution
function F(x). Calculate the realization of sample mean, sample variance, sample standard
deviation and construct a graph of the sample distribution function.
Problem 2
The test results from two subjects of 8 randomly drawn students are available:
Student's number 1 2 3 4 5 6 7 8
Score in the 1st test 80 50 36 58 42 60 56 68
Score in the 2nd test 65 60 35 39 48 44 48 61
Calculate and interprete the sample correlation coefficient. To make calculation easier use
the following sums:
8
i=1
xi = 450,
8
i=1
yi = 400,
8
i=1
x2
i = 26 684,
8
i=1
y2
i = 20 836,
8
i=1
xiyi = 23 214,
Problem 3
Let X1, . . . , Xn be a random sample from distribution with the expected value  and the
variance 2
. Find the expected value and the variance of the sample mean and the expected
value of the sample variance.
Problem 4
Consider a large year-class at university, taking exams from statistics. Assume, that its test
score follows the normal distribution with expected value 72 and the standard deviation
9. Find the probability, that:
a) the score of randomly drawn student will be greater then 80 points.
b) the mean of the score of 10 randomly drawn students will be greater then 80 points.
Problem 5
Let X1, . . . , Xn be a random sample from a distribution with a distribution function F(x),
n  2. For any fixed real number x calculate the expected value and the variance of the
sample distribution function Fn(x) = 1
n
card{i; Xi  x}.
Problem 6
Derive a joint probability function of a random sample from an alternative (Bernoulli)
distribution A().