1. seminar
Problem 1
Let X  N(0, 1)  (x). Prove the formula (-u) = 1 - (u)
[Greek capital letter  stands for distribution function of standard normal random varia-
ble.]
Problem 2
Find the probability that the random variable X  N(20, 16) takes the value less than 12
or greater than 28.
Problem 3
Let X1, X2 be independent random variables, Xi  N(0, 1), i = 1, 2. Determine the
distribution of transformed random variable Y = 3 + X1 - 2X2, further determine its
parameters and finally determine its first quartile value (25% quantile).
Problem 4
The random variable X1  N(19, 0.52
) represents the exchange rate of dolar and X2 
N(32, 0.62
) represents the exchange rate of euro. Correlation R(X1, X2) = -0.8. What is
the probability that the value of currency basket 0.65X1 + 0.35X2 will be greater than 24?
Clues on how:
X1
X2
 N2(
19
32
,
0.25 -0.072
-0.072 0.36
0.65X1 + 0.35X2 = (0.65 0.35)
X1
X2
Problem 5
Let X1, X2, X3, X4 be independent random variables , Xi  N(0, 1),
i = 1, 2, 3, 4. Determine the distribution of transformed random variable X = X1

3
X2
2 +X2
3 +X2
4
?
Problem 6
Let X  F(n1, n2). Determine the distribution of transformed random variable Y = 1
X
Problem 7
Let X  t(n). Determine the distribution of transformed random variable Y = X2
Problem 8
Prove that
a) u = -u1b)
t(n) = -t1-(n)
c) F(n1, n2) = 1
F1-(n2,n1)
Problem 9
sbírka 10.1,10.2,10.3,10.5 - Looking up in probability tables