MATHEMATICS
List of exercises 6
Probability and Stochastic processes
1. We toss two coins. What is the probability of getting had on one coin and tail on the
another one?
Remark: 300 years ago mathematicians thought the answer is 1/3.
2. Throwing a dice (1-6). What is the probability of
a) getting a number divisible by 2 and by 3.
b) getting a number divisible by 2 or by 3.
c) getting a number divisible by 2, if we know it is divisible by 3.
3.
- a casino game, where players may choose to place bets on either a single number
or a range of numbers, the colors red or black, or whether the number is odd or even, or
if the numbers. To determine the winning number and color, a croupier spins a wheel in
one direction, then spins a ball in the opposite direction and the ball eventually loses falls
onto the wheel and into one of 37 pockets (numbered 0-36).
What is your possible and average prot if you play fot 100zª and
Roulette
a) you bet on a particuar number. In case you win, you money is multiplied by 36.
b) you bet on odd numbers. In case you win, you money is multiplied by 2.
How do you understand the answer?
4.
There are only two groups in a school: A nad B. In the group A 75%
out of 4 girls and 50% out of 16 boys have passed the subject Mathematics. In the group
B the numbers were 30% out of 15 girls and 20% out 5 boys. What is the chance of passing
Mathematics for a
Simpson's paradox.
a) random girl,
b) random boy,
from that school?
5.
- A game, where there were 3 boxes and a prize only in one of them.
A player chooses on box and then the host shows which of the another boxes is empty.
After that a player has one last chance to change their mind and choose the other (from
two remaining) box. What should they do?
Precisely, asssume that the player chose box no. 1. and then box no. 3 turned out to
be empty (E3 ). Let P1 denotes that the prize is in box no.1 Calculate P(P1 |E3 ) and
P(P2 |E3 ).
Let's make a deal
6. We throw with 2 (distinguishable) dices. Let X denote the greater from obtained numbers.
Calculate P(X = 1), P(X = 2), ..., P(X = 6) and P(X ≤ 5). What is the average greater
number E[X].
7. Tossing a coin. Let X be the rst moment when we got head. Calculate P(X = 5),
P(X > 1).
8. Let X be of Poisson distribution with parameter λ = 2.
Calculate P(X = 2) and P(X ≥ 1).
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9. Show that time of waiting for a signal T (exponential distribution) in signal process has
a property of memorylessness, i.e.
P(T > t + s|T > s) = P(t > t)
10. Let Nt be the Poisson process with parameter λ = 2. Calculate
• P(Nt < 2), P(Nt ≤ 2), P(Nt > 1),
• P(N1/2 = 3 ∩ N1 = 2), P(Nt = 1 ∩ N2t = 1),
• P(N2 = 3|N1 = 1), P(Nt+s = n + 1|Nt = n).
11. Calculate E [Nt ], where Nt is the Poisson process with parameter λ > 0.
12. Calculate variance of Brownian motion Bt
Z
∞
x2 kt (x)dx.
−∞
Try to interpret it using 3σ property for normal distribution.
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