Introduction To Probability

 PROBABILITY AND RANDOM SIGNALS

INTRODUCTION TO PROBABLITY:

Probability is a number that describes a set.

▪ The higher the number, the more probability there is.

▪ In this sense probability is like a quantity that measures a physical

phenomenon

▪ Methods used for assigning probabilities

▪ Personal Approach

▪ Relative Frequency Approach

▪ Classical Approach

▪ Study of probability refers to an experiment consisting of a procedure and

observations.

▪ In random variables, each observation corresponds to one or more

numbers.

▪ In stochastic processes, each observation corresponds to a function of time.

▪ The word stochastic means random.

▪ In stochastic /random processes, we study random functions of time.

▪ Almost all practical applications of probability involve multiple observations

taken over a period of time.

SET THEORY:

Set theory is the mathematical theory of well-determined collections,

called sets, of objects that are called members, or elements, of the set.

• Set Theory is a branch of mathematics that investigates sets and their

properties.

• The mathematical basis of probability is the theory of sets

• Basic sets

▪ Operations

▪ Venn Diagram

▪ Properties

▪ Conditional probability

▪ Independent events

▪ Baye's formula

▪ Discrete and continuous random variables

▪ Distributions 

▪ Density functions

PROBABILITY DISTRIBUTIONS

▪ Binomial Distribution 

▪ Poisson Distribution

▪ Hyper geometric Distribution 

▪ Normal Distribution 

▪ Uniform Distribution 

▪ Exponential Distribution

STATISTICS:

• Statistics is the science concerned with developing and studying methods

for collecting, analyzing, interpreting and presenting empirical data.

• Statistics studies methodologies to gather, review, analyze and draw 

conclusions from data.

▪ Mean

▪ Variance

▪ Standard deviations

▪ Moments and 

▪ Moment generating functions

REGRESSION AND CURVE FITTING:

Regression takes a group of random variables and tries to find a

mathematical relationship between them.

Curve fitting is the process of specifying the model that provides the best fit to the curve in your data.

▪ Linear regression 

▪ Curve fitting

▪ Limits theorems

STOCHASTIC PROCESSES:

• A stochastic process is a system which evolves in time while

undergoing chance fluctuations.

▪ First and second order characteristics 

▪ Applications