Probability theory
Randomness is the essence of communication. Communication theory involves the assumption that the transmitter is connected to a source, whose output, the receiver is not able to predict with certainty.Thus, probability plays a major role in communication.
We shall introduce some of the basic concepts of probability theory by defining some terminology relating to random experiments
Outcome
The end result of an experiment. For example, if the experiment consists
of throwing a die, the outcome would be anyone of the six faces,
Random experiment
An experiment whose outcomes are not known in advance. (e.g. tossing a
coin, throwing a die)
Random event
A random event is an outcome or set of outcomes of a random experiment
that share a common attribute. For example, considering the experiment of
throwing a die, an event could be the 'face F1 ' or 'even indexed faces'
(F2 , F4 , F6 ).
Sample space
The sample space of a random experiment is a mathematical abstraction
used to represent all possible outcomes of the experiment.
Conditional probability
Conditional probability denoted P (B | A) ; it represents the probability of B occurring, given that A has occurred. To give a simple example, let a bowl contain 3 resistors and 1 capacitor. The occurrence of the event 'the capacitor on the second draw' is very much dependent on what has been drawn at the first instant.
We shall introduce some of the basic concepts of probability theory by defining some terminology relating to random experiments
Outcome
The end result of an experiment. For example, if the experiment consists
of throwing a die, the outcome would be anyone of the six faces,
Random experiment
An experiment whose outcomes are not known in advance. (e.g. tossing a
coin, throwing a die)
Random event
A random event is an outcome or set of outcomes of a random experiment
that share a common attribute. For example, considering the experiment of
throwing a die, an event could be the 'face F1 ' or 'even indexed faces'
(F2 , F4 , F6 ).
Sample space
The sample space of a random experiment is a mathematical abstraction
used to represent all possible outcomes of the experiment.
Conditional probability
Conditional probability denoted P (B | A) ; it represents the probability of B occurring, given that A has occurred. To give a simple example, let a bowl contain 3 resistors and 1 capacitor. The occurrence of the event 'the capacitor on the second draw' is very much dependent on what has been drawn at the first instant.
Random variable
Cumulative density function
The cumulative distribution function (CDF), or just distribution function, evaluated at 'x', is the probability that a real-valued random variable X will take a value less than or equal to x. In other words, CDF(x) = Pr(X≤x), where Pr denotes probability.
Probability density function (PDF)
Probability density function or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value
Where Fx is the cdf
- When the value of a variable is the outcome of a statistical experiment, that variable is a random variable.
- An example will make this clear. Suppose you flip a coin two times. This simple statistical experiment can have four possible outcomes: HH, HT, TH, and TT.
- Now, let the variable X represent the number of Heads that result from this experiment. The variable X can take on the values 0, 1, or 2. In this example, X is a random variable; because its value is determined by the outcome of a statistical experiment.
Cumulative density function
The cumulative distribution function (CDF), or just distribution function, evaluated at 'x', is the probability that a real-valued random variable X will take a value less than or equal to x. In other words, CDF(x) = Pr(X≤x), where Pr denotes probability.
Probability density function (PDF)
Probability density function or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value
Where Fx is the cdf