Npdf of the product of two gaussian random variables

The product of two gaussian random variables is not gaussian distributed. Similarly to the scalar case, the pdf of a gaussian random vector is completely characterized by its. Let x be a continuous random variable with pdf f x definite and positive in a. The pdf gives the distribution of a sample covariance. Characteristic functions of the product of two gaussian random variables and the product of a gaussian and a gamma random. Lets consider independent gaussian variables for a moment. This letter considers the distribution of product for two correlated real gaussian random variables with nonzero means and arbitrary variances, which arises widely in radar and communication societies.

Distribution of the sum of a complex gaussian and the. A state of the art am lcar oliveira 2,3teresa oliveira antonio seijasmac as 1,3 1department of economics. Continuous random variables and probability density functions probability density functions properties examples expectation and its properties the expected value rule linearity. It is essentially a chi distribution with two degrees of freedom. The mean of the product of correlated normal random variables arises in many areas. Related to the product distribution are the ratio distribution, sum distribution see list of convolutions of probability distributions and difference distribution. Why is the product of two normal random variables not. Universidade da coruna spain 2department of sciences and technology.

In probability theory and statistics, the multivariate normal distribution, multivariate gaussian distribution, or joint normal distribution is a generalization of the onedimensional normal distribution to higher dimensions. On products of gaussian random variables zeljka stojanac 1, daniel suessy, and martin klieschz2 1institute for theoretical physics, university of cologne, germany 2 institute of theoretical physics and astrophysics, university of gdansk, poland may 29, 2018 sums of independent random variables form the basis of many fundamental theorems. This paper uses integral calculus to derive the pdf of the product of two nonidentical, independent, and triangularly distributed random variables. Product of two gaussian pdf s with different dimensions. On the product of independent complex gaussians ieee. The distribution of product of two normally distributed variables come.

Is the product of two gaussian random variables also a. If several random variable are jointly gaussian, the each of them is gaussian. To obtain the probability density function pdf of the product of two continuous random variables r. Products of normal, beta and gamma random variables. How do we find the joint pdf of the product of two. Products and convolutions of gaussian probability density. Gaunta,b athe university of manchester buniversity of oxford abstract. Note that the product of two normal random variables is not normal, but the product of their pdfs is proportional to the pdf of another normal. Why is the product of two normal random variables not normal, but.

V ariance of the product of two gaussian variables. Correlation in random variables suppose that an experiment produces two random variables, x and y. One of the best ways to visualize the possible relationship is to plot the x,ypairthat is produced by several trials of the experiment. We consider a bivariate distribution of the two variables.

Sums of discrete random variables 289 for certain special distributions it is possible to. It is possible to use this repeatedly to obtain the pdf of a product of multiple but xed number n2 of random variables. Product of two gaussian pdfs is a gaussian pdf, but product of two gaussian variables is not gaussian. T o derive a formula for the vari ance, you apply the ito expansion and order of magnitude analysis to the two components. Pdf products and ratios of two gaussian class correlated. Characteristic functions of the product of two gaussian random. A rayleigh distribution is often observed when the overall magnitude of a vector is related to its directional components.

Products of independent random variables have already been studied for more than 50 years 5 but are still subject of ongoing research 69. Our approach is based upon the fact that the real or imaginary component of the product of two independent complex gaussian random variables has a laplace marginal probability density function pdf. Is the product of two gaussian random variables also a gaussian. The distribution of the product of two random variables which have lognormal distributions is again lognormal. They proved that such pdf has the same properties of the. Assume that the random variable x has support on the interval a. The product of correlated normal samples case was recently addressed by nadarajaha and pogany. In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to any.

Products and ratios of two gaussian class correlated weibull random variables. On the product of two correlated complex gaussian random. This distribution is useful in many problems, for example radar and communication systems. This is itself a special case of a more general set of results where the logarithm of the product can be written as the sum of the logarithms. The distribution of the product of two triangular random. Two random variables clearly, in this case given f xx and f y y as above, it will not be possible to obtain the original joint pdf in 16. Variance of product of multiple random variables cross. Take the product of the two density functions, group the arguments of the exponentials in the most straightforward way, define a new variable with the form z 2. The intuition which i use is that for two random variables, we need two independent streams of randomness, which we then mix to get the right correlation structure. The related problem of characterizing products of independent random variables seems to be much more challenging.

In fact, the most recent work on the properties of the sum of two independent ggrv is given in 10, where zhao et al. Take a look at the central limit theorem central limit theorem wikipedia when you multiply two pdfs of two random variables, say x and y, you are taking the convolution of the two curves. In probability theory, a normal or gaussian or gauss or laplacegauss distribution is a type of continuous probability distribution for a realvalued random variable. Also, the product space of the two random variables is assumed to fall entirely in the rst quadrant. A simple technique to reduce the correlated case to the uncorrelated is to diagonalize the system. Pdf mean and variance of the product of random variables. The probability density function of sum of a complex gaussian and the product of two complex gaussians is derived. Please see this link for the multiplication of two independent gaussian random variables.

In this paper, we derive the joint amplitude, phase distribution of the product of two independent nonzeromean complex gaussian random variables. The product of two random variables is a random variable and it is not possible to calculate the joint probability distribution of a single variable. What this gives you is a pdf that is the same as the pdf. You may recall that the product of a gaussian variable with itself follows a chisquared distribution, for example. It should be noted that this result is not the pdf of the product of two gaussian random variates. On the product of independent complex gaussians abstract. In this paper, we extend steins method to products of independent beta, gamma, generalised gamma and mean zero normal random variables. Stein operators and distributional theory robert e.

Simply knowing that the result is gaussian, though, is enough to allow one to predict the parameters of the density. In particular, it was shown that the probability density function of a product of certain independent and identically distributed iid random variables from the exponential family can be written in terms of meijer gfunctions 10. Product of two gaussian pdfs is a gaussian pdf, but. As a byproduct, we also derive closedform expressions for the exact pdf of the mean z. In this work, we investigate such products of normal random variables, products of their absolute values, and products. So, when i talk about multiplying two grvs and their product being a gaussian, i mean to say that the pdf of product of these two grvs will also be resemble a gaussian function. At each time the distribution of its possible positions in space will be a gaussian because the total displacement is the sum of a bunch of gaussiandistributed displacements, and the sum of gaussian variables is gaussian. X and y are said to be jointly normal gaussian distributed, if their joint pdf has the following form. Norma gaussian random variables important in the theory of probability central limit theorem prevalent in applications.

We say that x and y have a bivariate gaussian pdf if the joint pdf of x and y is given by f x y s x y x y 21 1 exp 2 1. Assume that all the separate channels undergo rayleigh fading, which means. In my understanding a gaussian random variable grv is a random variable with pdf resembling a gaussian function. We determine the exact probability density function pdf in terms of an infinite sum of modified bessel functions of second kind. On the distribution of the product of correlated normal. For the special case of two gaussian probability densities. Jointly gaussian random variablesjointly gaussian random variables let x and y be gaussian random variables with means. Distribution of the product of two normal variables. Arkadiusz gives the answer in the case of two independent gaussians. The fact that the means and variances add when summing s. The parameter is the mean or expectation of the distribution and also its median and mode. New results on the sum of two generalized gaussian. On the product of independent complex gaussians request pdf.

Moment generating function of the inner product of two. A ratio distribution also known as a quotient distribution is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. This distribution occurs in wireless communications where gaussian. One definition is that a random vector is said to be kvariate normally distributed if every linear combination of its k components has a univariate normal. For the two nonnegative random variables, we also compute the moment generating functions in terms of meijer gfunctions, and consequently. Given two usually independent random variables x and y, the distribution of the random variable z that is formed as the ratio z xy is a ratio. As a by product, we derive the exact distribution of the mean of the product of correlated normal random variables. The product is one type of algebra for random variables.

Exact distribution for the product of two correlated. More generally, one may talk of combinations of sums, differences, products and ratios. What is the distribution of a random variable that is the product of the two normal random variables. In this letter, we derive the exact joint probability density function pdf of the amplitude and phase of the product of two correlated nonzero mean complex gaussian random variables with arbitrary variances. That is, if two random variables are jointly gaussian, then uncorelatedness and independence are equivalent. The product of two independent normal variables is. In probability theory and statistics, the rayleigh distribution is a continuous probability distribution for nonnegativevalued random variables. We solve a problem that has remained unsolved since 1936 the exact distribution of the product of two correlated normal random variables. The variance can be found by transforming from two unit variance zero mean uncorrelated variables u, v. Sums of independent random variables form the basis of many fundamental theorems in probability theory and statistics, and therefore, are well understood. The general form of its probability density function is. An example of correlated samples is shown at the right. Here is a plot of the histogram of the product of t.

For the special case of two gaussian probability densities, the product density has mean and variance given by next prev up top index jos index jos pubs jos home search. Given random variables,, that are defined on a probability space, the joint probability distribution for, is a probability distribution that gives the probability that each of, falls in any particular range or discrete set of values specified for that variable. What is the distribution of a random variable that is the. What is the probability distribution function for the. But the product of two gaussian pdfs is a gaussian pdf. The theory and practice of obtaining the distribution of the product of two random variables have been addressed by donahue 1964, springer and thompson 1966, rohatgi 1976 and springer 1979.