Probability Calculus Anna Janicka
lecture III, 3.11.2020
INDEPENDENCE OF EVENTS BERNOULLI PROCESS
POISSON THEOREM
INTRODUCTION TO RANDOM VARIABLES
Plan for today
1. Independence of events 2. The Bernoulli Process
3. Approximation of the Bernoulli Process for large n – Poisson Theorem
4. Introduction to random variables
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)
3. Examples of good and not so good approximations
Random variables – basics
1. Motivation – functions of the results of an experiment
2. Definition of a random variable
3. Examples
◼ number of heads
◼ sum of points on dice
◼ the distance to a given point
Random variables – distribution
4. Functions of random variables
5. Examples of descriptions of random variables.
6. Definition of a random v. distribution
7. Different r.v. have the same distributions
notation: X ~
we forget about
Random variables – examples
8. Examples of random variables
◼ die roll
◼ discrete distributions
◼ Binomial distribution
◼ Geometric distribution
◼ Poisson distribution
◼ uniform distribution over an interval: a continuous distribution
◼ another continuous distribution
