Rayleigh Distribution

STATISTICS FOR DATA SCIENCE

LA 1

RAYLEIGH DISTRIBUTION

SUBMITTED BY:

PRANJUL NEMA

1NT20IS113

ABSTRACT

The Rayleigh distribution is a continuous probability distribution that describes the magnitude of a two dimensional vector in polar coordinates. It is often used to model the magnitude of noise in various engineering and scientific applications.

The probability density function (PDF) of the Rayleigh distribution is given by:

f(x) = x * exp (-x^2 / (2 * sigma^2)) / sigma^2

where x is the magnitude of the vector and sigma is a scale parameter. The mean and variance of the distribution are given by:

mean = sqrt (pi / 2) * sigma

variance = (4 - pi) / 2 * sigma^2

The Rayleigh distribution is a special case of the Weibull distribution, with a shape parameter of 2. It is also related to the chi-squared distribution, with one degree of freedom.

The Rayleigh distribution is often used to model the magnitude of noise in communication systems, such as the fading of a radio signal. It is also used in statistical hypothesis testing and in the analysis of wind speeds and wave heights.

INTRODUCTION

The Rayleigh distribution is a continuous probability distribution named after the English Lord Rayleigh. It is a special case of the Weibull distribution with a scale parameter of 2. When a Rayleigh is set with a shape parameter (σ) of 1, it is equal to a chi square distribution with 2 degrees of freedom. The notation X Rayleigh(σ) means that the random variable X has a Rayleigh distribution with shape parameter σ. The probability density function (X > 0) is:

pdf

Where e is Euler's number.

Variance and Mean (Expected Value) of a Rayleigh Distribution

The expected value (the mean) of a Rayleigh is:

EV

How this equation is derived involves solving an integral, using calculus:

The expected value of a probability distribution is:

E(x) = ∫ xf(x)dx.

Substituting in the Rayleigh probability density function, this becomes the improper integral:

Rayleigh expected value

Rayleigh expected value

Where:

  • exp is the exponential function,

  • d xis the differential operator.

    Solving the integral for you gives the Rayleigh expected value of σ √(π/2)The variance of a Rayleigh distribution is derived in a similar way, giving the variance formula of:

    Var(x) = σ2((4 – π)/2).

REAL LIFE APPLICATIONS

The Rayleigh distribution is a widely useddistribution in many fields, due to its simplicity and versatility in modelling the magnitude of noise or other continuous variables that are naturally expressed in terms of vector magnitudes. Some real-life applications of the Rayleighdistribution include

  1. Communication systems: The Rayleigh distribution is often used to model the magnitude of noise in communication systems, such as the fading of a radio signal.

  2. Statistical hypothesis testing: The Rayleigh distribution is commonly used in statistical hypothesis testing, particularly in the analysis of windspeeds and wave heights.

  3. Engineering: The Rayleigh distribution is used in various engineering applications,such as the analysis of structural reliability and the prediction of fatigue life in materials.

    PROBLEM AND SOLUTION

    The Rayleigh distribution is a continuous probability distribution that describes the magnitude of a complex-valued random variable whose real and imaginary parts are independently and identically distributed according to the normal distribution. It is often used to model the amplitude of a signal in the context of wireless communication or the strength of a wind speed.

    One problem that may arise when working with the Rayleigh distribution is determining the appropriate parameters for the distribution. The shape of the distribution is determined by a single parameter called the scale parameter, which determines the spread of the distribution. Estimating the scale parameter can be difficult, especially if the data is noisy or limited. One solution to this problem is to use maximum likelihood estimation or other statistical techniques to estimate the scale parameter fromthe data.

    Another problem that may arise is the need to perform statistical tests or make statistical inferences about the data. For example, you may want to test whether the data is consistent with a Rayleigh distribution or compare the scale parameters of two different datasets. Inthese cases, it may be necessary to use statistical tests such as the Anderson-Darling test or the Kolmogorov-Smirnov test to evaluate the goodnessof fit of the data to the Rayleigh distribution.

    Finally, it is also important to consider the assumptions of the Rayleigh distribution when using it to model data. The Rayleigh distribution assumes that the real and imaginary parts of the complex-valued random variable are independently and identically distributed according to the normal distribution. If these assumptions are not met, the Rayleigh distribution may not be an appropriate model for the data. In this case, it may be necessary to use a different distribution or transform the data in some way to meet the assumptions of the Rayleigh distribution.

    FORMULAES

    REFERENCES

    https://web.stanford.edu/class/archive/cs/cs109/cs109.1166/stuff/reviewSoln.pdf

    https://www.tutorialspoint.com/statistics/required_sample_size.htm

    https://en.wikipedia.org/wiki/Rayleigh_distribution