Probability Calculus Anna Janicka
lecture III, 22.10.2019
INDEPENDENCE OF EVENTS BERNOULLI PROCESS
POISSON THEOREM
Plan for today
1. Independence of events 2. The Bernoulli Process
3. Approximation of the Bernoulli Process for
large n – Poisson Theorem
Independence of Events
1. Definition
2. Examples
die roll
choosing a card
Symmetric.
Stochastic independence
Independence of Events – cont.
3. Independence of 3+ events
4. Examples.
The definition may not be simplified!
Independence and pairwise independence
Independence of Events – cont. (2)
5. Theorem. Independence conditions
Bernoulli Process
1. Definition
a finite or an infinite process
Bernoulli Process – cont.
2. Examples
3. Probability in a Bernoulli process:
probability of exactly k successes in n trials
Bernoulli Process – cont. (2)
n=10
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45
0 1 2 3 4 5 6 7 8 9 10
p=0,1 p=0,25 p=0,5 p=0,75 p=0,9
Bernoulli Process – cont. (3)
n=11
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35
0 1 2 3 4 5 6 7 8 9 10 11
p=1/6 p=0,3 p=0,5 p=0,7 p=5/6
Bernoulli Process – cont. (4)
4. Examples
coin flip die roll
5. The most probable number of successes
6. Infinite sequence of heads
Poisson Theorem
1. Poisson Theorem
2. Assessment of approximation error
Poisson Theorem – cont.
The Poisson and Bernoulli processes
0 0,05 0,1 0,15 0,2 0,25 0,3
0 1 2 3 4 5 6 7 8 9 10
n=10, p=0,3 n=30, p=0,1 n=60, p=0,05 n=100, p=0,03 n=300, p=0,01 lambda=3
Poisson Theorem – cont. (2)
