Mumbai University > Electronics and Telecommunication Engineering > Sem 5 > Random Signal Analysis

Marks: 10M

Year: Dec 2015

• Poisson distribution:

  Definition: The probability distribution of the Poisson random variable X, representing the number of outcomes occurring in a given time interval or specified region represented as t, is P(X=x)=($(e^λ)^t$.$(λt)^x$)/x!, x=0,1,2……

Where λ is the average number of outcomes per unit time or region.

Poison distribution is a limiting form of binomial distribution when n→∞,p→0 and np remains constant and μ'=np.

Mean of Poisson distribution (E(X)):

E(X)=λt

Variance of Poisson distribution (Var(X)):

Var(X)=λt

Properties:

• The number of outcomes occurring during a time interval is independent of the number that occurs in any other disjoint time interval. So, it is memoryless.

• The probability that a single outcome will occur during a very short time intervsl is proportional to the length of the time interval. It doesn’t depend on the number of outcomes that occur outside this time interval.

• The probability that more than one outcome will occur in such a short time interval is negligible.

Applications:

• The number of alpha particles emitted by a radioactive source in time interval t

• The number of telephone calls received at a particular telephone exchange in the time interval t.

• Rayleigh Distribution:

  Definition: A random variable is said to have a Rayleigh distribution with parameter if its p.d.f. is given by the probability law,

f(x)= $xe^{{-x^2}/2σ²}$ /σ², x>0, σ>0

or, f(x)=$2(x-a)e^{{(a-x)^2}/b}$ /b, x ≥ a (parameters a and b)

(Originally derived by Lord Rayleigh (J.W. Strutt) in the field of acoustics.)

The graph below shows various Rayleigh distributions. The distribution [Rayleigh(1)], is sometimes referred to as the standard Rayleigh distribution.

Mean: E(X )= α√(π/2)

Var(X) = (2-π/2)$α^2$

Applications:

The Rayleigh (1) distribution is frequently used to model wave heights in oceanography, and in communication theory to describe hourly median and instantaneous peak power of received radio signals. It has been used to model the frequency of different wind speeds over a year at wind turbine sites.

Rayleigh distribution arises when studying the magnitude of a complex number whose real and imaginary parts both follow a zero-mean Gaussian distribution. It arises often in the study of non-coherent communication systems and also in the study of land mobile communication channels, where the phenomenon known as fading is often modelled using Rayleigh random variables. In communication systems, the signal amplitude values of a randomly received signal usually can be modelled using Rayleigh random variables. - Gaussian Distribution:

Definition: A continuous random variable X, with parameters m and σ^2 is normal or Gaussian if it has a probability density function.

$f_x$(x) = N(m, $α^2$)= $e^{(x-m)^2}$/ α√2π.2$α^2$, $-\infty $ < x < $\infty $

The distribution of a normal random variable with mean zero and variance 1 is called a standard normal distribution N(0,1).

Mean=Median=Mode=m

Variance=σ^2,Standard deviation=σ

Moment Generating Function $M_X$(t)= $e^{tm+({t^2 σ^2}/2)}$

Properties:

• The curve is symmetric about a vertical axis through the mean m.

• The mode which is the value of the random variable for which $f_X$ (x) is maximum, occurs at x=m.

• The normal curve approaches asymptotically the horizontal axis as x increases in either direction away from the mean.

• The curve has its point of inflection atx=m±σ and is concave downward if m-σ < X < m+σ and is concave upward otherwise.

Areas of Application

The most important continuous probability distribution in the statistic field is normal distribution. It describes many phenomena that occur in nature like rainfall and meteorological studies; in industry, in error calculations of experiments, statistical quality control, radar applications and in research.

Under certain conditions, normal distribution provides a good continuous approximation to the binomial and hyper geometric distribution.