is then & = \iint\limits_{\{(x,y): x + y \le z\}} f_{X}(x) f_{Y}(y) \ \text{d}y \ \text{d}x v ( , Asking for help, clarification, or responding to other answers. {\displaystyle X,Y\sim {\text{Norm}}(0,1)} n 1 = To learn more, see our tips on writing great answers. {\displaystyle x} ( The product of non-central independent complex Gaussians is described by ODonoughue and Moura[13] and forms a double infinite series of modified Bessel functions of the first and second types. ) 1 , 1 with y . t , ( {\displaystyle h_{X}(x)=\int _{-\infty }^{\infty }{\frac {1}{|\theta |}}f_{x}\left({\frac {x}{\theta }}\right)f_{\theta }(\theta )\,d\theta } Y Note that {\displaystyle f_{y}(y_{i})={\tfrac {1}{\theta \Gamma (1)}}e^{-y_{i}/\theta }{\text{ with }}\theta =2} It is a theorem in Multivariate Probability that two linear functions AU and BU of a Gaussian vector U are independent if an only if A.var (U).B^T = 0. ( ; 1 [ z {\displaystyle f_{X}(\theta x)=\sum {\frac {P_{i}}{|\theta _{i}|}}f_{X}\left({\frac {x}{\theta _{i}}}\right)} ( & = \int_{\mathbb{R}} \int_{-\infty}^{z} f_{X}(x) f_{Y}(y - x) \ \text{d}y \ \text{d}x \\ \mathbb{P}(XY \le k) ( f Here $D$ is the region in the first quadrant which is "below" the hyperbola $xy=z$. is their mean then. {\displaystyle f_{X}} Note that when $-20\lt v \lt 20$, $\log(20/|v|)$ is. i It is possible to use this repeatedly to obtain the PDF of a product of multiple but xed number (n>2) of random variables. {\displaystyle z=x_{1}x_{2}} Could anyone please indicate a general strategy (if there is any) to get the PDF (or CDF) of the product of two random variables, each having known distributions and limits? i u f Contents 1 Algebra of random variables 2 Derivation for independent random variables 2.1 Proof [3] 2.2 Alternate proof 2.3 A Bayesian interpretation Thus, by the above formula, for any $0 < x \leq 1$, Can the product of a Beta and some other distribution give an Exponential? ) where p = t T, and random variable Z has pdf f ( z): (source: tri.org.au) Next, X 2 has pdf, say f 2 ( x 2): (source: tri.org.au) The desired product of random variables is: Y = X 1 X 2 = { 0 with probability p Z X 2 with probability 1 p Then, the mixed pdf of Y is: pdf ( Y) = { p if y = 0 ( 1 p) h ( z x 2) if y 0 f 2 This reduces to the fact that the joint probability (or probability density) function of X and Y "splits" as a product: z i are two independent random samples from different distributions, then the Mellin transform of their product is equal to the product of their Mellin transforms: If s is restricted to integer values, a simpler result is, Thus the moments of the random product c therefore has CF Hi, I am working on this question here, which asks to find the probability from a joint pdf with two random variables. ) y its CDF is, The density of x X W ( and. = 1 f Let X and Y be continuous random variables with joint PDF fX,Y ( x, y ). z z Thus there may be a sort of closed form for your density function. t The product of correlated Normal samples case was recently addressed by Nadarajaha and Pogny. {\displaystyle u_{1},v_{1},u_{2},v_{2}} Does a purely accidental act preclude civil liability for its resulting damages? d are uncorrelated, then the variance of the product XY is, In the case of the product of more than two variables, if ( rev2023.3.17.43323. {\displaystyle \sum _{i}P_{i}=1} asymptote is ) First of all, letting Since the variance of each Normal sample is one, the variance of the product is also one. s Which holomorphic functions have constant argument on rays from the origin? is. ) t i X is normal distributed and Y is chi-square distributed. {\displaystyle g} Please help. What's the point of issuing an arrest warrant for Putin given that the chances of him getting arrested are effectively zero? whose moments are, Multiplying the corresponding moments gives the Mellin transform result. we have, High correlation asymptote Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. ( Y \end{align*}, $\partial g = \begin{bmatrix} 1/u & -t/u^2\\ 0 & 1\end{bmatrix}$, $$ f_{T,U}(t,u) = f_{X,Y}(g(t,u)) \cdot | \mathrm{det}\,\partial g | = f_X(t/u) \cdot f_Y(u) \,/\, |u|.$$, $$ f_{X\cdot Y}(t) = \int_{\mathbb{R}} f_{T,U}(t,u) \, \partial u = \int_{-\infty}^\infty f_X\left(\frac{t}{u}\right) \cdot f_Y(u) \, \frac{\partial u}{u}. 0 where W is the Whittaker function while \mathbb{P}(XY \le k) Do the inner-Earth planets actually align with the constellations we see? L11.9 The PDF of a Function of Multiple Random Variables. 2 further show that if Assume that the random variable X has support on the interval (a;b) and the random variable Y has support on the in-terval (c;d). h r [ be independent samples from a normal(0,1) distribution. Here $f_U (u) = 1$, $0 < u <1$, $F_V (v) = v$, $0 < v < 1$, and $F_V (v) = 1$, $v \geq 1$. Chapter. ) y with X,Y independent r.v. Download as PDF Printable version Languages On this Wikipedia the language links are at the top of the page across from the article title. y {\displaystyle (1-it)^{-1}} t $$h(v) = \int_{y=-\infty}^{y=+\infty}\frac{1}{y}f_Y(y) f_X\left (\frac{v}{y} \right ) dy$$. Why is there no video of the drone propellor strike by Russia, Create a simple Latex macro which expands the format to sequence, A challenge between Sandman and Lucifer Morningstar. z = ) ! The PDF of V = XY is f V (v)= f X,Y x, v x 1 |x| d x. p Y = d The random variable M is an example. h x Norm In particular, an indicator ( }, The variable For example to record the height and weight of each person in a community or 2 For $z \in \mathbb{R}$ we have x ) Indicator random variables are closely related to events. [16] A more general case of this concerns the distribution of the product of a random variable having a beta distribution with a random variable having a gamma distribution: for some cases where the parameters of the two component distributions are related in a certain way, the result is again a gamma distribution but with a changed shape parameter.[16]. & = \int_{\mathbb{R}} \int_{-\infty}^{z} f_{X}(x) f_{Y}(y - x) \ \text{d}y \ \text{d}x \\ Contradiction in derivatives as linear approximations. , t ) . a The pdf gives the distribution of a sample covariance. {\displaystyle x_{t},y_{t}} ) {\displaystyle x\geq 0} Moment generating function technique. ) | n Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. ) Y ; In your derivation, you do not use the density of $X$. 2 y X Why is geothermal heat insignificant to surface temperature? , Comments. above is a Gamma distribution of shape 1 and scale factor 1, ( ( ) | x x ) \mathbb{P}(X + Y \le z) Modified 2 years, 1 month ago. x thus. Theorem 2.1 Let ( X, Y) denote a bivariate normal random vector with zero means, unit variances and correlation coefficient . x ( Let $Z=XY$. It doesn't look like uniform. {\displaystyle \varphi _{Z}(t)=\operatorname {E} (\varphi _{Y}(tX))} 2 ( ( = Var 2 I know what you mean informally, but formally $P(U = u) = 0$ since $U$ is continuous so $P(UV\leq x \mid U = u)$ does not make sense. Observe $g(T,U) = (X,Y)$ where $g(t,u) := (t/u, u)$. = = EDIT: Here's a particularly simple example. Since $X\sim\mathcal{U}(0,2)$, $$f_X(x) = \frac{1}{2}\mathbb{I}_{(0,2)}(x)$$so in your convolution formula \end{align*} {\displaystyle W_{2,1}} y Theorem. q ( {\displaystyle \Gamma (x;k_{i},\theta _{i})={\frac {x^{k_{i}-1}e^{-x/\theta _{i}}}{\Gamma (k_{i})\theta _{i}^{k_{i}}}}} If the characteristic functions and distributions of both X and Y are known, then alternatively, z x The expected value of the product of two random variables. $|Y|$ is ten times a $U(0,1)$ random variable. and Finally, find the density function of $\exp(U)$. x Y ) Thus, making the transformation The pdf of a function can be reconstructed from its moments using the saddlepoint approximation method. | ) The shaded area within the unit square and below the line z = xy, represents the CDF of z. I have attempted the question here, but I think that my answer is wrong, considering that the value I got for the probability exceeds 1, when it should be between 0 and 1. Let $X$ and $Y$ be independent random variables with $\mathbb{P}(Y=0) = 0$. i from the definition of correlation coefficient. = My particular need is the following: Let $w :=u \cdot v$. ) ) f z \mathbb{P}(X + Y \le z) K {\displaystyle n} f_{XY}(z)dz &= 0\ \text{otherwise}. It only takes a minute to sign up. z z The product of two independent Gamma samples, ( 1 is a product distribution. 1 x ) = & = \iint\limits_{\{(x,y): x + y \le z\}} f_{X}(x) f_{Y}(y) \ \text{d}y \ \text{d}x & = \boxed{\int_{-\infty}^{\ln(k)} \int_{\mathbb{R}} f_{\ln(Z)}(x) f_{\ln(Y)}(y-x) \ \text{d}x \ \text{d}y.} z {\displaystyle X,Y} Let $X$ ~ $U(0,2)$ and $Y$ ~ $U(-10,10)$ be two independent random variables with the given distributions. A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. {\displaystyle |d{\tilde {y}}|=|dy|} & = \boxed{\int_{-\infty}^{\ln(k)} \int_{\mathbb{R}} f_{\ln(Z)}(x) f_{\ln(Y)}(y-x) \ \text{d}x \ \text{d}y.} {\displaystyle \theta X\sim h_{X}(x)} ( X For instance, to obtain the pdf of $XY$, begin with the probability element of a $\Gamma(2,1)$ distribution, $$f(t)dt = te^{-t}dt,\ 0 \lt t \lt \infty.$$, Letting $t=-\log(z)$ implies $dt = -d(\log(z)) = -dz/z$ and $0 \lt z \lt 1$. | {\displaystyle \rho } ) f What's the PDF of $w$? f i y Y z x ( Values within (say) $\varepsilon$ of $0$ arise in many ways, including (but not limited to) when (a) one of the factors is less than $\varepsilon$ or (b) both the factors are less than $\sqrt{\varepsilon}$. Further, the density of Gamma distributions with the same scale parameter are easy to add: you just add their shape parameters. | {\displaystyle y={\frac {z}{x}}} 4 t s $$ In this case the ) ) Z | @nth I agree with the comment by Therkel, this answer is not accurate, $f_U(u) \neq p(U=u)$. How should I respond? {\displaystyle h_{X}(x)} | {\displaystyle \delta p=f(x,y)\,dx\,|dy|=f_{X}(x)f_{Y}(z/x){\frac {y}{|x|}}\,dx\,dx} X 1 d Y {\displaystyle \theta X} ( Y . ) 2 Then use the usual "convolution" formula to find the density function of $U$, where $U=\ln X +\ln Y$. How can I stay longer in my flight stop cities without much additional flight cost? 1 Added: To my surprise, an integral not far from what is necessary for the first approach can be expressed in terms of modified Bessel functions and modified Struve functions (whatever those are). = The convolution of A $\Gamma(1,1)$ plus a $\Gamma(1,1)$ variate therefore has a $\Gamma(2,1)$ distribution. z {\displaystyle p_{U}(u)\,|du|=p_{X}(x)\,|dx|} t If X, Y are drawn independently from Gamma distributions with shape parameters x f | Setting f This example illustrates the case of 0 in the support of X and Y and also the case where the support of X and Y includes the endpoints . then, from the Gamma products below, the density of the product is. y , x ) r = The conditional density is = $$ ) {\displaystyle {_{2}F_{1}}} - , we have $$ = | / ) and this extends to non-integer moments, for example. What would be the formal approach? = W The APPL code to find the distribution of the product is. f The Stack Exchange reputation system: What's working? log i 27 Author by Balerion_the_black. . d Y n X e {\displaystyle x} ) Expected value of product of independent random variables - Probability Theory, Statistics and . | ( n , 2 While the pdf describes the RV, we already see that they are slightly different mathematical concepts. are independent variables. Let Z= XYa product of two normally distributed random variables, we consider the distribution of the random variable Z. 2 x are two independent, continuous random variables, described by probability density functions , {\displaystyle s} h(v) &= \frac{1}{40} \int_{-10}^{0} \frac{1}{|y|} \mathbb{I}_{0\le v/y\le 2}\text{d}y+\frac{1}{40} \int_{0}^{10} \frac{1}{|y|}\mathbb{I}_{0\le v/y\le 2}\text{d}y\\ &= \frac{1}{40} \int_{-10}^{0} \frac{1}{|y|} \mathbb{I}_{0\ge v/2\ge y\ge -10}\text{d}y+\frac{1}{40} \int_{0}^{10} \frac{1}{|y|}\mathbb{I}_{0\le v/2\le y\le 10}\text{d}y\\&= \frac{1}{40} \mathbb{I}_{-20\le v\le 0} \int_{-10}^{v/2} \frac{1}{|y|}\text{d}y+\frac{1}{40} \mathbb{I}_{20\ge v\ge 0} \int_{v/2}^{10} \frac{1}{|y|}\text{d}y\\ = | $$ Furthermore, for the . {\displaystyle dx\,dy\;f(x,y)} d X u 2 Thus, in cases where a simple result can be found in the list of convolutions of probability distributions, where the distributions to be convolved are those of the logarithms of the components of the product, the result might be transformed to provide the distribution of the product. \begin{align*} Products often are simplified by taking logarithms. The Mellin transform of a distribution The distribution of the product of non-central correlated normal samples was derived by Cui et al. 2 However this approach is only useful where the logarithms of the components of the product are in some standard families of distributions. f {\displaystyle \theta } m What are the benefits of tracking solved bugs? is a function of Y. 1 2 n If all three coins match, then M = 1; otherwise, M = 0. n 2 ) = {\displaystyle z=e^{y}} ( , follows[14], Nagar et al. f | 2 The purpose of this one is to derive the same result in a way that may be a little more revealing of the underlying structure of $XY$. This page titled 4.2: Probability Distribution Function (PDF) for a Discrete Random Variable is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. x 2 X What is the difference between \bool_if_p:N and \bool_if:NTF. Intuition behind product distribution pdf, Probability distribution of the product of two dependent random variables, Identifying lattice squares that are intersected by a closed curve. {\displaystyle \varphi _{X}(t)} Theorem 2.1 derives the exact PDF of the product of two correlated normal random variables. x are samples from a bivariate time series then the x = On this Wikipedia the language links are at the top of the page across from the article title. p {\displaystyle z=yx} X Here's another way using convolution and the functional equation of the natural logarithm, provided $X,Y \ge 1$ almost surely. and The product is one type of algebra for random variables: Related to the product distribution are the ratio distribution, sum distribution (see List of convolutions of probability distributions) and difference distribution. i 1 , v 2 Let P implies = \int_{\mathbb{R}} \int_{-\infty}^{z - x} f_{X}(x) f_{Y}(y) \ \text{d}y \ \text{d}x \\ Hence the PDF of $UV$ is given, for $0 < x < 1$, by ( e corresponds to the product of two independent Chi-square samples They are completely specied by a joint pdf fX,Y such that for any event A (,)2, P{(X,Y . 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