Products and ratios of two gaussian class correlated weibull random variables petros s. Let x be a continuous random variable with pdf f x definite and positive in a, b, with. On the product of independent complex gaussians ieee. The problem is, they wont be that easy to deal with either, because the result then depends on if the variances of u and v are the same or not, or if the means of u and v are zero. If several random variable are jointly gaussian, the each of them is gaussian. Products and ratios of two gaussian class correlated. Generalizations to more than two variables can also be made. Based on this results, the probability density function pdf and the cumulative distribution function cdf of the sum distribution are obtained. Products of normal, beta and gamma random variables. Thanks for contributing an answer to cross validated. Also, the product space of the two random variables is assumed to fall entirely in the rst quadrant.
Products and ratios of two gaussian class correlated weibull. An example of correlated samples is shown at the right. Furthermore, a useful union upper bound for the distribution of the sum of two correlated weibull variates is also presented. The above ideas are easily generalized to two or more random variables. The argument in the linked document goes like this.
May 12, 2008 if the two random variables x and y are independent, then the pdf of z xy is probably i havent done the calculation a bessel function. Product of two gaussian pdfs with different dimensions. New results on the sum of two generalized gaussian. 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. An evalued random variable x is gaussian if the real valued random variable hx,x. Let z xya product of two normally distributed random variables, we consider the distribution of the random variable z. On the product of two correlated complex gaussian random. The pdf of the product of two independent random variables, x and the gaussian variable y. On computing distributions of products of 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. Nov 28, 2017 sums of independent random variables form the basis of many fundamental theorems in probability theory and statistics, and therefore, are well understood.
The pdf of a product of two independent rvs is their. The parameter is the mean or expectation of the distribution and also its median and mode. The probability density distribution is discussed for two correlated random variables based on an approximation to a normal gaussian law using hermite polynomials of two variables. On the product of independent complex gaussians abstract. Take the product of the two density functions, group the arguments of the exponentials in. Product of two gaussian pdfs is a gaussian pdf, but product of two gaussian variables is not gaussian 2 probability density function and the minimal sufficient statistics for. Equivalently, in the frequency domain, their characteristic functions multiply. Here is a plot of the histogram of the product of t. The continuous version of the joint pmf is called the joint pdf.
We call this new distribution the complex double gaussian distribution. The problem is, they wont be that easy to deal with either, because the result then depends on if the variances of u and v are the same or not, or if. Integrating these ordinary differ ential equations you get analytical expressions fo r the expectation and vari ance. To add to my confusion, i found a note by bromiley, where it is argued that the product of two normal independent random variables is a normal variable again which i thought was not the case. In this paper, we derive the joint amplitude, phase distribution of the product of two independent nonzeromean complex gaussian random variables. Characteristic functions of the product of two gaussian random. On products of gaussian random variables request pdf. Correlation in random variables suppose that an experiment produces two random variables, x and y. Take and two gaussian random variables with mean and variance. In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to any. Products of independent random variables have already been studied for more than 50 years 5 but are still subject of ongoing research 69.
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 how to cite this work order a printed hardcopy comment on. The product of two normal variables might be a nonnormal distribution. The two conditions for the form method to be accurate, i. Gaussian random variable an overview sciencedirect topics. 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. For the special case of two gaussian probability densities. 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. Lets consider independent gaussian variables for a moment. The general form of its probability density function is. The product of two complex gaussian random variables is a problem that has. Pdf the distribution of the product of two triangular. This distribution is useful in many problems, for example radar and communication systems.
In this work, we investigate such products of normal random variables, products of their absolute values, and products of their. The product of two normal variables might be a nonnormal distribution skewness is 2 p 2. If the two random variables x and y are independent, then the pdf of z xy is probably. Assume that all the separate channels undergo rayleigh fading, which means. In this section we consider only sums of discrete random variables, reserving the case of continuous random variables for. To be consistent with the analysis presented in section 3. In probability theory, a normal or gaussian or gauss or laplacegauss distribution is a type of continuous probability distribution for a realvalued random variable.
That is, if two random variables are jointly gaussian, then uncorelatedness and independence are equivalent. It does not say that a sum of two random variables is the same as convolving those variables. If u and v are both normally distributed, then the product will be a linear combinations of two chisquared random variables. Moreover, we express the moment generating functions of the two. Sums of independent random variables form the basis of many fundamental theorems in probability theory and statistics, and therefore, are well understood. Why is the product of two normal random variables not normal. The distribution of the product of two triangular random. First approaches to this question are considered in 5, authors conclusions is that distribution function of a product of two independent normal variables is proportional to a bessel function of the second kind of a purely. Is the product of two gaussian random variables also a gaussian. X and y are said to be jointly normal gaussian distributed, if their joint pdf. We consider a bivariate distribution of the two variables. 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. A basic result from the theory of random variables is that when you sum two independent random variables, you convolve their probability density functions pdf. Pdf products and ratios of two gaussian class correlated.
X is normal distributed and y is chisquare distributed. Pdf mean and variance of the product of random variables. Why is the product of two normal random variables not. Product of two normal distributions math help forum. Assume that the random variable x has support on the interval a. Distribution of the product of two normal variables. Cross validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. And the given references dont say the product is a pdf either. Product of two gaussian pdfs is a gaussain pdf, but produt of two.
Is the product of two gaussian random variables also a. The cf of the sum of two independent gg random variables is then deduced. It says that the distribution of the sum is the convolution of the distribution of the individual. Product of two gaussian pdfs is a gaussian pdf, but product of two gaussian variables is not gaussian. Product of two gaussian pdfs is a gaussian pdf, but product of two gaussian variables is not gaussian asked 5 years, 3 months ago 31 the product of two gaussian random variables is not gaussian distributed. Mar 06, 2019 if u and v are both normally distributed, then the product will be a linear combinations of two chisquared random variables. Mac as product two normal variables september, 2018521. This paper uses integral calculus to derive the pdf of the product of two nonidentical, independent, and triangularly distributed random variables. As an initial application, we employ this new tool to accurately and efficiently compute the probability density function pdf of the product of independent random variables. We introduce a new approximate multiresolution analysis mra using a single gaussian as the scaling function, which we call gaussian mra gmra. A sum of gaussian random variables is a gaussian random.
One of the best ways to visualize the possible relationship is to plot the x,ypairthat is produced by several trials of the experiment. Similar to covariance, the correlation is a measure of the linear relationship between random variables. Lest this seem too mysterious, the technique is no different than pointing out that since you can add two numbers with a calculator, you can add n numbers with the same. Complex double gaussian pdf for the special cases where one or both.
On the product of two correlated complex gaussian random variables abstract. New results on the sum of two generalized gaussian random. On the product of independent complex gaussians request pdf. If k is diagonal matrix, then x 1 and x 2 are independent case 1 and case 2. Karagiannidis3 1 electrical and computer engineering department, university of patras, rion, 26500 patras, greece. This contradiction problem results in a significant inaccuracy of the results delivered by the combined approach involving the form method. Please see this link for the multiplication of two independent gaussian random variables. You may recall that the product of a gaussian variable with itself follows a chisquared distribution, for example. Identity implies that and the product measure 1 n have the same. We consider the typical case of two random variables that are either both discrete or both continuous. How to calculate the product between gaussian and exponential.
The related problem of characterizing products of independent random variables seems to be much more challenging. The product of two gaussian random variables is not gaussian distributed. In the case of the multivariate gaussian density, the argument ofthe exponential function. For the two nonnegative random variables, we also compute the moment generating functions in terms of meijer gfunctions, and consequently. That the sum of two independent gaussian random variables is gaussian follows. The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. X and y are said to be jointly normal gaussian distributed, if their joint pdf has the following form. A sum of gaussian random variables is a gaussian random variable. Product of two gaussian pdfs is a gaussian pdf, but.
Simply knowing that the result is gaussian, though, is enough to allow one to predict the parameters of the density. On the distribution of the product of correlated normal. Applying such maps to properly normalized rankone tensor results in a product of dgaussian random variables. E much of the theory of banach spacevalued gaussian random variables depends on a fundamental integrability result due to fernique. Gaussian variables and a linear failure surface, cannot be satisfied simultaneously with the selected pair of random variables. In cases where one variable is discrete and the other continuous, appropriate modifications are easily made. A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions.