Probability Tutorial

Reference: http://web.lums.edu.pk/~ihsan/teaching.html

Probability Basics:

1. Sample Space: Set of all possible outcomes

2. Probability Distribution Law:It assigns a probability to each measurable subset (Event) of possible outcomes of a random experiment. It is required to satisfy three probability axioms.

3. Event: Subset of Sample Space

4. Conditional Probability --> Theorem of Total Probability --> Bayes Rule ---> Application: Gunshot Detection System (see attached M-file)

5. Independence

Random Variable: It is a function that assigns a numerical value to each possible outcome of the experiment.

RVs:

1. Discrete R.V.: A random variable is called discrete if its range is finite (e.g. X=Number of heads in n coin tosses or {0,1,2,...,n}) or countably infinite (e.g. Y=Number of tosses before the first tail or {0,1,2,...,infinity}).

- A discrete random variable has an associated probability mass function (PMF), which gives the probability of each numerical value that the random variable can take. Notation: p_X(x)=P({X=x})

-PMF a) The Bernoulli R.V. --> The Binomial R.V. [Application: Capacity Provisioning in LAN] --> The Poisson R.V. [Application: Optical Communication System] b) The Geometric R.V.

-Expectations & Variance of above R.V.s

-Joint PMFs and Conditional PMFs --> CHAIN RULE