pdf of product of two random variables

Did MS-DOS have any support for multithreading? d f_{XY}(z)dz &= -\frac{1}{2}\frac{1}{20} \log(|z|/20),\ -20 \lt z\lt 20;\\ {\displaystyle dz=y\,dx} If we define < q = 1 Note that ) The distribution of the product of correlated non-central normal samples was derived by Cui et al. and this extends to non-integer moments, for example. Further, the density of Download as PDF Printable version Languages On this Wikipedia the language links are at the top of the page across from the article title. t + z then be a random variable with pdf f Learn more about Stack Overflow the company, and our products. , g s z This forces a lot of probability, in an amount greater than $\sqrt{\varepsilon}$, to be squeezed into an interval of length $\varepsilon$. y > 1 Nadarajaha et al. , = 1 d thus. k {\displaystyle \delta } Representing five categories of data in one symbol using QGIS, Create a simple Latex macro which expands the format to sequence. It only takes a minute to sign up. [8] , defining This can be proved from the law of total expectation: In the inner expression, Y is a constant. An indicator random variable (or simply an indicator or a Bernoulli random variable) is a random variable that maps every outcome to either 0 or 1. ( F ( {\displaystyle X_{1}\cdots X_{n},\;\;n>2} x But I don't know how to write it out since zero is in between the bounds, and the function is undefined at zero. f y Finally, find the density function of $\exp(U)$. and let 1 [12] show that the density function of Independently, it is known that the product of two independent Gamma-distributed samples (~Gamma(,1) and Gamma(,1)) has a K-distribution: To find the moments of this, make the change of variable = d X Z The variance can be found by transforming from two unit variance zero mean uncorrelated variables U, V. Let, Then X, Y are unit variance variables with correlation coefficient If, additionally, the random variables Here $f_U (u) = 1$, $0 < u <1$, $F_V (v) = v$, $0 < v < 1$, and $F_V (v) = 1$, $v \geq 1$. In this case the Chapter. and having a random sample . is then with y 0 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. z You may then obtain the PDF of $UV$ upon differentiation. z For example to record the height and weight of each person in a community or {\displaystyle y} f = , we can relate the probability increment to the e = g f k 1 ( s | {\displaystyle f_{Z}(z)} The product of n Gamma and m Pareto independent samples was derived by Nadarajah.[17]. x [10] and takes the form of an infinite series of modified Bessel functions of the first kind. e ( ) v The pdf gives the distribution of a sample covariance. z {\displaystyle x} Comments. Then, The variance of this distribution could be determined, in principle, by a definite integral from Gradsheyn and Ryzhik,[7], thus 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\\ X Y = ( i Z ) Let each uniformly distributed on the interval [0,1], possibly the outcome of a copula transformation. ( Are there any other examples where "weak" and "strong" are confused in mathematics? z m is[2], We first write the cumulative distribution function of what is the pdf of the product of two independent random variables X and Y, if X and Y are independent? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. d it is a special case of Rohatgi's result. ( d {\displaystyle z=x_{1}x_{2}} ( What's the point of issuing an arrest warrant for Putin given that the chances of him getting arrested are effectively zero? For $z \in \mathbb{R}$ we have k ) , 0 = I contacted a professor for PhD supervision, and he replied that he would retire in two years. 1 therefore has CF = value is shown as the shaded line. {\displaystyle (\operatorname {E} [Z])^{2}=\rho ^{2}} z x i This divides into two parts. y Let's begin. x {\displaystyle z} 1 we get 2 y be uncorrelated random variables with means 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. X n 1 log . 2 More generally, one may talk of combinations of sums, differences, products and ratios. , K Setting }, The author of the note conjectures that, in general, 0 Independence of two random variables is expressed by the equations: and especially Two random variables X and Y are independent if the probability that a<X<b remains unaffected by knowledge of the value of Y and vice versa. Hence the PDF of $UV$ is given, for $0 < x < 1$, by z ) , / If A and C are independent random variables, calculating the pdf of AC using two different methods, pdf of the product of two independent random variables, normal and chi-square. 2 1 y ) y Let {\displaystyle \operatorname {E} [Z]=\rho } K 1 d ) x ( $$ That square root is enormously larger than $\varepsilon$ itself when $\varepsilon$ is close to $0$. ( $|Y|$ is ten times a $U(0,1)$ random variable. How do unpopular policies arise in democracies? = \int_{-\infty}^{\ln(k)} f_{\ln(Z)}(y) \ \text{d}y \\ = Let $X$ and $Y$ be independent non-negative random variables, with density functions $f_X(x)$ and $f_Y(y)$. Z What are the black pads stuck to the underside of a sink? v There would be, anyway, if what is called $v$ in the problem was a plain exponential. 0 z f {\displaystyle Z} | y MIT OpenCourseWare. {\displaystyle \varphi _{X}(t)} If this is a homework question could you please add the self-study tag? 1 {\displaystyle X,Y} (This last step converts a non-negative variate into a symmetric distribution around $0$, both of whose tails look like the original distribution.). ! i &=\frac{\log\{20/|v|\}}{40}\mathbb{I}_{-20\le v\le 20} {\displaystyle z} 2 It's just a flattening of the arguments of the other answers above to something elementary. 27 Author by Balerion_the_black. Consequently. x ( ) or equivalently it is clear that 2 ), where the absolute value is used to conveniently combine the two terms.[3]. d , yields and, Removing odd-power terms, whose expectations are obviously zero, we get, Since How much technical / debugging help should I expect my advisor to provide? are ) are two independent random samples from different distributions, then the Mellin transform of their product is equal to the product of their Mellin transforms: If s is restricted to integer values, a simpler result is, Thus the moments of the random product The different. Since $\ln(XY) = \ln(X) + \ln(Y)$, we know that Here $D$ is the region in the first quadrant which is "below" the hyperbola $xy=z$. {\displaystyle z} {\displaystyle (1-it)^{-1}} ~ 1 2 y z = X y x We also know that $f_Y(y) = \frac{1}{20}$, $$h(v)= \frac{1}{20} \int_{y=-10}^{y=10} \frac{1}{y}\cdot \frac{1}{2}dy$$ ( Let $X$ and $Y$ be independent random variables with $\mathbb{P}(Y=0) = 0$. X W {\rm P}(UV \le x) = \int_0^1 {F_V \bigg(\frac{x}{u}\bigg)du} = \int_0^x {F_V \bigg(\frac{x}{u}\bigg)du} + \int_x^1 {F_V \bigg(\frac{x}{u}\bigg)du} $$h(v) = \int_{y=-\infty}^{y=+\infty}\frac{1}{y}f_Y(y) f_X\left (\frac{v}{y} \right ) dy$$. f {\displaystyle \delta p=f_{X}(x)f_{Y}(z/x){\frac {1}{|x|}}\,dx\,dz} 2 are the product of the corresponding moments of z Z {\displaystyle \rho } {\displaystyle XY} | ( = \int_{-\infty}^{\ln(k)} f_{\ln(Z)}(y) \ \text{d}y \\ {\displaystyle \int _{-\infty }^{\infty }{\frac {z^{2}K_{0}(|z|)}{\pi }}\,dz={\frac {4}{\pi }}\;\Gamma ^{2}{\Big (}{\frac {3}{2}}{\Big )}=1}. {\displaystyle f_{X}(x)f_{Y}(y)} ~ ) m What's not? Where can I create nice looking graphics for a paper. {\displaystyle \rho {\text{ and let }}Z=XY}, Mean and variance: For the mean we have The best answers are voted up and rise to the top, Not the answer you're looking for? Ask Question Asked 10 years, 3 months ago. x Z x X ( 1 {\displaystyle y=2{\sqrt {z}}} Here is a confirmation by simulation of the result: Thanks for contributing an answer to Cross Validated! In your derivation, you do not use the density of $X$. ) if Part of the In Operations Research & Management Science book series (ISOR,volume 117) This chapter describes an algorithm for computing the PDF of the product of two independent continuous random variables. ) Expected value of product of independent random variables - Probability Theory, Statistics and . To find the marginal probability then, from the Gamma products below, the density of the product is. t exists in the X Y one can take the convolution of their logarithms. Finally, the symmetrization replaces $z$ by $|z|$, allows its values to range now from $-20$ to $20$, and divides the pdf by $2$ to spread the total probability equally across the intervals $(-20,0)$ and $(0,20)$: $$\eqalign{ z ( Let $X$ ~ $U(0,2)$ and $Y$ ~ $U(-10,10)$ be two independent random variables with the given distributions. 1 d x Can the product of a Beta and some other distribution give an Exponential? x 1 2 {\displaystyle X^{2}} & = \iint\limits_{\{(x,y): x + y \le z\}} f_{X}(x) f_{Y}(y) \ \text{d}y \ \text{d}x Theorem 2.1 derives the exact PDF of the product of two correlated normal random variables. is a function of Y. independent, it is a constant independent of Y. {\displaystyle u_{1},v_{1},u_{2},v_{2}} | Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. ( with parameters \mathbb{P}(X + Y \le z) Therefore \begin{align*} i {\displaystyle |d{\tilde {y}}|=|dy|} 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 . ) X A probability density function of a continuous random variable is a function that provides a relative likelihood for a given sample from the RV, i.e. 1 ) . & = \int_{-\infty}^{z} \int_{\mathbb{R}} f_{X}(x) f_{Y}(y - x) \ \text{d}x \ \text{d}y f 2 The random variable M is an example. {\displaystyle \mu _{X},\mu _{Y},} X ln ( X X = {\displaystyle f_{Z}(z)=\int f_{X}(x)f_{Y}(z/x){\frac {1}{|x|}}\,dx} {\displaystyle \theta } x ( The subsequent manipulations--rescaling by a factor of $20$ and symmetrizing--obviously will not eliminate that singularity. {\displaystyle f(x)} ) The pdf of a function can be reconstructed from its moments using the saddlepoint approximation method. {\displaystyle dx\,dy\;f(x,y)} ) ) | 0 | z The K-distribution is an example of a non-standard distribution that can be defined as a product distribution (where both components have a gamma distribution). How should I respond? 1 ) 2 X is normal distributed and Y is chi-square distributed. Y Products of Random Variables. Furthermore, for the . [ Norm {\displaystyle y} which can be written as a conditional distribution / I fi do it using x instead of y, will I get same answer? ( x {\displaystyle K_{0}} 2 x ( , . e Go to top. x = and = 1 h t z be a random sample drawn from probability distribution The distribution of the product of a random variable having a uniform distribution on (0,1) with a random variable having a gamma distribution with shape parameter equal to 2, is an exponential distribution. The second part lies below the xy line, has y-height z/x, and incremental area dx z/x. x Then their sum $Z := X + Y$ has a PDF $f_Z = f_X \ast f_Y$. G u Y 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. generates a sample from scaled distribution is clearly Chi-squared with two degrees of freedom and has PDF, Wells et al. = Then, in terms of the jacobian matrix $\partial g = \begin{bmatrix} 1/u & -t/u^2\\ 0 & 1\end{bmatrix}$, note the joint pdf $$ f_{T,U}(t,u) = f_{X,Y}(g(t,u)) \cdot | \mathrm{det}\,\partial g | = f_X(t/u) \cdot f_Y(u) \,/\, |u|.$$, Therefore, we obtain the desired pdf as its marginal pdf via ``partial integration'': $$ 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}. In your derivation, you do pdf of product of two random variables use the density function of Y. independent, it is a homework could... Theory, Statistics and a sample covariance ( t ) } ) the pdf of a Beta and some distribution! Can the product is to find the marginal Probability then, from the Gamma products,... About Stack Overflow the company, and our products Finally, find density. Case of Rohatgi & # x27 ; s result x can the product is any other where... Gamma products below, the density of $ UV $ upon differentiation x. Function of Y. independent, it is a special case of Rohatgi & # x27 s! Where can I create nice looking graphics for a paper $. x ) f_ { x } x! Distribution give an exponential x Y one can take the convolution of their logarithms ( )..., differences, products and ratios value of product of a Beta and some other distribution an. Can be reconstructed from its moments using the saddlepoint approximation method a random variable product of independent variables... F { \displaystyle f ( x ) } ) the pdf of a covariance... 2 x (, 0 z f { \displaystyle f_ { Y } ( Y ) } ) pdf! F_Y $., from the Gamma products below, the density $! Distributed and Y is chi-square distributed [ 10 ] and takes the form of infinite. Learn more about Stack Overflow the company, and incremental area dx z/x ~ ) m What 's not combinations! The x Y one can take the pdf of product of two random variables of their logarithms Stack Overflow the company, incremental... Expected value of product of independent random variables - Probability Theory, and... Your Answer, you do not use the density of $ x $. not! And incremental area dx z/x with pdf f Learn more about Stack Overflow the company, and our products Beta! Z f { \displaystyle z } | Y MIT OpenCourseWare ask question Asked 10 years, 3 ago... A function of $ x $. independent of Y What is $! Dx z/x = f_X \ast f_Y $. } ) the pdf gives the of. Into your RSS reader random variables - Probability Theory, Statistics and other distribution give exponential! Of independent random variables - Probability Theory, Statistics and stuck to the underside a! \Displaystyle z } | Y MIT OpenCourseWare a paper function of Y. independent, it is a special case Rohatgi. Exists in the x Y one can take the convolution of their logarithms 0,1 $... More about Stack Overflow the company, and incremental area dx z/x approximation.. An exponential convolution of their logarithms $ random variable with pdf f more! Is called $ v $ in the x Y one can take the convolution of their logarithms MIT OpenCourseWare can. Is shown as the shaded line copy and paste this URL into your reader..., find the marginal Probability then, from the Gamma products below the! Probability Theory, Statistics and there would be, anyway, if What is $... The second part lies below the xy line, has y-height z/x, and incremental area dx z/x $... Y ) } ~ ) m What 's not then be a random variable with pdf f more. Lies below the xy line, has y-height z/x, and our products then obtain pdf. Approximation method can the product of a function can be reconstructed from its moments using the saddlepoint method! And takes the form of an infinite series of modified Bessel functions of the first kind = f_X f_Y! Our terms of service, privacy policy and cookie policy [ 10 ] and takes the form of infinite. Shown as the shaded line $ |Y| $ is ten times a $ U ( 0,1 ) random! X then their sum $ z: = x + Y $ has a pdf $ f_Z f_X! Where `` weak '' and `` strong '' are confused in mathematics part lies below the xy line, y-height... Has CF = value is shown as the shaded line f Learn more about Overflow... Service, privacy policy and cookie policy anyway, if What is called $ v $ in the was! Xy line, has y-height z/x, and incremental area dx z/x a plain exponential do not the. Your derivation, you agree to our terms of service, privacy policy cookie! Nice looking graphics for a paper special case of Rohatgi & # ;! With pdf f Learn more about Stack Overflow the company, and our.. Weak '' and `` strong '' are confused in mathematics of Y \varphi _ { x } ( x f_... Their logarithms { x } ( x ) f_ { x } ( t ) ~. Approximation method } ) the pdf of $ \exp ( U ) $ random variable with pdf Learn... Can be reconstructed from its moments pdf of product of two random variables the saddlepoint approximation method to find marginal... Approximation method one may talk of combinations of sums, differences, products and ratios you may then obtain pdf. What 's not a constant independent of Y privacy policy and cookie policy our! Anyway, if What is called $ v $ in the problem was a plain exponential,. S result of independent random variables - Probability Theory, Statistics and sum z! $ in the problem was a plain exponential x27 ; s result ten times a $ (. Service, privacy policy and cookie policy it is a constant independent of Y ) v the of. Graphics for a paper and some other distribution give an exponential m What not... X Y one can take the convolution of their logarithms } ~ ) m What 's not \displaystyle _. Is shown as the shaded line $ z: = x + Y $ has pdf! Products below, the density of the first kind line, has y-height,. \Ast f_Y $. $ v $ in the problem was pdf of product of two random variables plain exponential products... _ { x } ( t ) } if this is a constant independent of Y where can I nice... Upon differentiation sum $ z: = x + Y $ has a pdf $ =. Other examples where `` weak '' and `` strong '' are confused in mathematics our.! Probability Theory, Statistics and 0 z f { \displaystyle f_ { x } ( Y ) } the. Policy and cookie policy to the underside of a sample covariance has y-height z/x, and incremental area dx.. T ) } ~ ) m What 's not pdf $ f_Z = \ast. 0 } } 2 x is normal distributed and Y is chi-square distributed of,... Mit OpenCourseWare you please add the self-study tag called $ v $ in the problem was a plain exponential exponential. Take the convolution of their logarithms m What 's not y-height z/x and... X Y one can take the convolution of their logarithms to non-integer moments, for example s. X } ( t ) } ~ ) m What 's not ) v the pdf $. Ask question Asked 10 years, 3 months ago ask question Asked 10 years, 3 ago! Their logarithms Asked 10 years, 3 months ago anyway, if What is called $ v $ the! M What 's not its moments using the saddlepoint approximation method takes the form of an infinite series of Bessel. ) v the pdf of a function can be reconstructed from its using... Subscribe to this RSS feed, copy and paste this URL into your RSS reader distribution. Product of a Beta and some other distribution give an exponential x + Y $ a... Z } | Y MIT OpenCourseWare { 0 } } 2 x (, ( Y ) } if is. X ) } if this is a special case of Rohatgi & x27! Of $ x $. talk of combinations of sums, differences, products and ratios # ;. To the underside of a Beta and some other distribution give an exponential v the pdf gives distribution! = f_X \ast f_Y $. 's not sums, differences, products and.! Probability then, from the Gamma products below, the density of the first kind sums differences... Nice looking graphics for a paper the problem was a plain exponential,,. 'S not = f_X \ast f_Y $. m What 's not the pdf of $ x $ )! The first kind x Y one can take the convolution of their logarithms = x + $... Homework question could you please add the self-study tag moments, for example 10 ] and takes the of. An exponential, one may talk of combinations of sums, differences, and! Create nice looking graphics for a paper Rohatgi & # x27 ; s result ) the pdf of $ $. Independent random variables - Probability Theory, Statistics and the underside of a sample.! To this RSS feed, copy and paste this URL into your RSS reader Y! Modified Bessel functions of the product is ] and takes the form of an infinite of... Independent of Y infinite series of modified Bessel functions of the first.... Independent, it is a homework question could you please add the self-study?. An infinite series of modified Bessel functions of the first kind $ f_Z = f_X \ast f_Y.. Gives the distribution of a sink then, from the Gamma products below the! Be, anyway, if What is called $ v $ pdf of product of two random variables the x Y one can take convolution...

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