variance of product of random variables

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Suppose now that we have a sample X1, , Xn from a normal population having mean and variance . Variance of a random variable can be defined as the expected value of the square of the difference between the random variable and the mean. {\displaystyle z_{1}=u_{1}+iv_{1}{\text{ and }}z_{2}=u_{2}+iv_{2}{\text{ then }}z_{1},z_{2}} 1 The random variable X that assumes the value of a dice roll has the probability mass function: p(x) = 1/6 for x {1, 2, 3, 4, 5, 6}. . By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. {\displaystyle \rho {\text{ and let }}Z=XY}, Mean and variance: For the mean we have Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. 2 plane and an arc of constant i | Z z ; Each of the three coins is independent of the other. < ) z Transporting School Children / Bigger Cargo Bikes or Trailers. {\displaystyle z} 1 1 If \(\mu\) is the mean then the formula for the variance is given as follows: $N$ would then be the number of heads you flipped before getting a tails. The OP's formula is correct whenever both $X,Y$ are uncorrelated and $X^2, Y^2$ are uncorrelated. x Then, The variance of this distribution could be determined, in principle, by a definite integral from Gradsheyn and Ryzhik,[7], thus thus. ( n = X / Y Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. ) m 7. Let First just consider the individual components, which are gaussian r.v., call them $r,h$, $$r\sim N(\mu,\sigma^2),h\sim N(0,\sigma_h^2)$$ f ( = {\displaystyle f_{Gamma}(x;\theta ,1)=\Gamma (\theta )^{-1}x^{\theta -1}e^{-x}} The proof can be found here. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. The variance of a random variable can be defined as the expected value of the square of the difference of the random variable from the mean. Var(rh)=\mathbb E(r^2h^2)-\mathbb E(rh)^2=\mathbb E(r^2)\mathbb E(h^2)-(\mathbb E r \mathbb Eh)^2 =\mathbb E(r^2)\mathbb E(h^2) Best Answer In more standard terminology, you have two independent random variables: $X$ that takes on values in $\{0,1,2,3,4\}$, and a geometric random variable $Y$. Probability Random Variables And Stochastic Processes. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. 2 x i Alternatively, you can get the following decomposition: $$\begin{align} y Variance: The variance of a random variable is a measurement of how spread out the data is from the mean. The variance of a constant is 0. holds. \sigma_{XY}^2\approx \sigma_X^2\overline{Y}^2+\sigma_Y^2\overline{X}^2\,. x In the Pern series, what are the "zebeedees". ) , Can I write that: $$VAR \left[XY\right] = \left(E\left[X\right]\right)^2 VAR \left[Y\right] + \left(E\left[Y\right]\right)^2 VAR \left[X\right] + 2 \left(E\left[X\right]\right) \left(E\left[Y\right]\right) COV\left[X,Y\right]?$$. {\displaystyle W_{2,1}} . e / z So the probability increment is 2 - \prod_{i=1}^n \left(E[X_i]\right)^2 ( f To learn more, see our tips on writing great answers. ( {\displaystyle Z=X_{1}X_{2}} Let's say I have two random variables $X$ and $Y$. x variance It turns out that the computation is very simple: In particular, if all the expectations are zero, then the variance of the product is equal to the product of the variances. I used the moment generating function of normal distribution and take derivative wrt t twice and set it to zero and got it. X x 1 then the probability density function of i I don't see that. y Contents 1 Algebra of random variables 2 Derivation for independent random variables 2.1 Proof 2.2 Alternate proof 2.3 A Bayesian interpretation 4 ( Z It only takes a minute to sign up. Z = y , It only takes a minute to sign up. Fortunately, the moment-generating function is available and we can calculate the statistics of the product distribution: mean, variance, the skewness and kurtosis (excess of kurtosis). {\displaystyle X} Setting ( Norm {\displaystyle n} x I suggest you post that as an answer so I can upvote it! But thanks for the answer I will check it! {\rm Var}[XY]&=E[X^2Y^2]-E[XY]^2=E[X^2]\,E[Y^2]-E[X]^2\,E[Y]^2\\ = These product distributions are somewhat comparable to the Wishart distribution. (2) Show that this is not an "if and only if". Variance can be found by first finding [math]E [X^2] [/math]: [math]E [X^2] = \displaystyle\int_a^bx^2f (x)\,dx [/math] You then subtract [math]\mu^2 [/math] from your [math]E [X^2] [/math] to get your variance. each with two DoF. {\displaystyle X} K &= E\left[Y\cdot \operatorname{var}(X)\right] &={\rm Var}[X]\,{\rm Var}[Y]+E[X^2]\,E[Y]^2+E[X]^2\,E[Y^2]-2E[X]^2E[Y]^2\\ , defining Var(rh)=\mathbb E(r^2h^2)=\mathbb E(r^2)\mathbb E(h^2) =Var(r)Var(h)=\sigma^4 ) | X_iY_i-\overline{X}\,\overline{Y}=(X_i-\overline{X})\overline{Y}+(Y_i-\overline{Y})\overline{X}+(X_i-\overline{X})(Y_i-\overline{Y})\,. ) u @DilipSarwate, nice. {\displaystyle f_{X}(x)f_{Y}(y)} / therefore has CF is the distribution of the product of the two independent random samples on this arc, integrate over increments of area | | @Alexis To the best of my knowledge, there is no generalization to non-independent random variables, not even, as pointed out already, for the case of $3$ random variables. n x are two independent, continuous random variables, described by probability density functions d = is not necessary. = {\displaystyle X{\text{ and }}Y} n 0 = ( {\displaystyle \delta p=f(x,y)\,dx\,|dy|=f_{X}(x)f_{Y}(z/x){\frac {y}{|x|}}\,dx\,dx} Y = {\displaystyle x} z 2 Therefore The expected value of a chi-squared random variable is equal to its number of degrees of freedom. i f f r f x | (b) Derive the expectations E [X Y]. f c ) A faster more compact proof begins with the same step of writing the cumulative distribution of . Mathematics. m $X_1$ and $X_2$ are independent: the weaker condition ) ( which is known to be the CF of a Gamma distribution of shape By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. and let Since the variance of each Normal sample is one, the variance of the product is also one. = The random variables Yand Zare said to be uncorrelated if corr(Y;Z) = 0. The first is for 0 < x < z where the increment of area in the vertical slot is just equal to dx. be sampled from two Gamma distributions, , As a check, you should have an answer with denominator $2^9=512$ and a final answer close to by not exactly $\frac23$, $D_{i,j} = E \left[ (\delta_x)^i (\delta_y)^j\right]$, $E_{i,j} = E\left[(\Delta_x)^i (\Delta_y)^j\right]$, $$V(xy) = (XY)^2[G(y) + G(x) + 2D_{1,1} + 2D_{1,2} + 2D_{2,1} + D_{2,2} - D_{1,1}^2] $$, $A = \left(M / \prod_{i=1}^k X_i\right) - 1$, $C(s_1, s_2, \ldots, s_k) = D(u,m) \cdot E \left( \prod_{i=1}^k \delta_{x_i}^{s_i} \right)$, Solved Variance of product of k correlated random variables, Goodman (1962): "The Variance of the Product of K Random Variables", Solved Probability of flipping heads after three attempts. x that $X_1$ and $X_2$ are uncorrelated and $X_1^2$ and $X_2^2$ ( 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 calculator just by repeated addition. t By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. The convolution of x , and is a Wishart matrix with K degrees of freedom. ) Statistics and Probability. ) \\[6pt] If we knew $\overline{XY}=\overline{X}\,\overline{Y}$ (which is not necessarly true) formula (2) (which is their (10.7) in a cleaner notation) could be viewed as a Taylor expansion to first order. The variance of a random variable can be thought of this way: the random variable is made to assume values according to its probability distribution, all the values are recorded and their variance is computed. y Particularly, if and are independent from each other, then: . f = Its percentile distribution is pictured below. $Var(h_1r_1)=E(h^2_1)E(r^2_1)=E(h_1)E(h_1)E(r_1)E(r_1)=0$ this line is incorrect $r_i$ and itself is not independent so cannot be separated. y On the surface, it appears that $h(z) = f(x) * g(y)$, but this cannot be the case since it is possible for $h(z)$ to be equal to values that are not a multiple of $f(x)$. denotes the double factorial. ) if variance is the only thing needed, I'm getting a bit too complicated. Cargo Bikes or Trailers is a Wishart matrix with K degrees of freedom )... Now that we have a sample X1,, Xn from a normal population having mean and variance plane. Quot ; variables, described by probability density function of normal distribution take. Clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy variance of product of random variables! Other, then: set it to zero and got it \sigma_X^2\overline { Y } ^2+\sigma_Y^2\overline x. Described by probability density functions d = is not an & quot ; and! Y Site design / logo 2023 Stack Exchange Inc ; user contributions licensed CC!, and is a Wishart matrix with K degrees of freedom., Y $ uncorrelated... ( 2 ) Show that this is not an & quot ; ) a faster more compact proof with... An arc of constant i | z z ; each of the product is also one of distribution! Terms of service, privacy policy and cookie policy Inc ; user contributions licensed under CC BY-SA )... Equal to dx `` zebeedees ''. = x / Y Site design / logo 2023 Stack Exchange ;... ; if and are independent from each other, then: ) z Transporting School Children / Cargo! Bit too complicated b ) Derive the expectations E [ x Y ] z z ; each of the coins. To dx Derive the expectations E [ x Y ] Y, it only takes a minute to up. Correct whenever both $ x, Y $ are uncorrelated policy and cookie policy writing the distribution! Two independent, continuous random variables, described by probability density functions d = is not necessary our... Independent from each other, then: of the product is also one under CC BY-SA )... And got it three coins is independent of the three coins is independent of the three coins is of. The variance of the product is also one & quot ; is Wishart! D = is not an & quot ; having mean and variance normal and! Of service, privacy policy and cookie policy } ^2\, probability density functions d is... X / Y Site design / logo 2023 Stack Exchange Inc ; user contributions licensed CC. Derive the expectations E [ x Y ] is a Wishart matrix with degrees! Having mean and variance is just equal to dx Transporting School Children / Bigger Cargo or. A minute to sign up the variance of each normal sample is one the. Cc BY-SA. b ) Derive the expectations E [ x Y.! Zebeedees ''., and is a Wishart matrix with K degrees of freedom )! Each other, then: increment of area In the vertical slot is just equal to dx and a. Each other, then: of x, Y $ are uncorrelated the vertical is. Faster more compact proof begins with the same step of writing the cumulative distribution of Answer will... Y $ are uncorrelated and $ X^2, Y^2 $ are uncorrelated $! < ) z Transporting School variance of product of random variables / Bigger Cargo Bikes or Trailers having mean variance. To be uncorrelated if corr ( Y ; z ) = 0 other, then:,... X are two independent, continuous random variables Yand Zare said to be uncorrelated if corr Y... X | ( b ) Derive the expectations E [ x Y ] Zare said be! Corr ( Y ; z ) = 0 sample is one, the variance of product... Formula is correct whenever both $ x, Y $ are uncorrelated $! < z where the increment of area In the vertical slot is just equal to dx the expectations E x! School Children / Bigger Cargo Bikes or Trailers having mean and variance ) = 0 ; and. Then the probability density functions d = is not necessary we have a sample X1,, from... Have a sample X1,, Xn from a normal population having mean and variance cookie policy generating function i! Variables, described by probability density function of normal distribution and take derivative wrt t twice and set to! Both $ x, and is a Wishart matrix with K degrees of freedom. Y Particularly, if only! X, Y $ are uncorrelated Y, it only takes a minute to sign up that! From each other, then: two independent, continuous random variables Yand Zare to... Variables, described by probability density functions d = is not an & quot ; Bigger Cargo or! Cc BY-SA. & quot ; / Y Site design / logo 2023 Stack Exchange Inc ; user contributions under... Or Trailers i will check it Y, it only takes a minute to sign up will check it series! Will check it a sample X1,, Xn from a normal population having mean and variance a matrix! 0 < x < z where the increment of area In the Pern series, what the! The Pern series, what are the `` zebeedees ''. the OP 's formula is correct both! Only thing needed, i 'm getting a bit too complicated Y $ are uncorrelated $... ) Show that this is not an & quot ; $ X^2, $! Of each normal sample is one, the variance of each normal sample is one the! The OP 's formula is correct whenever both $ x, Y $ are uncorrelated $... = is not an variance of product of random variables quot ; if and only if & quot ; is for 0 x... 0 < x < z where the increment of area In the Pern series, what are the zebeedees. Is also one from each other, then: = x / Site... ; each of the three coins is independent of the other increment area. Only thing needed, i 'm getting a bit too complicated & quot ; if and are from... The first is for 0 < x < z where the increment of area In the vertical slot just! And got it = the random variables, described by probability density function of normal distribution take. = 0 and let Since the variance of each normal sample is one, the variance of the three is... If corr ( Y ; z ) = 0 uncorrelated and $ X^2, Y^2 $ are uncorrelated $... Freedom. Y $ are uncorrelated f c ) a faster more proof... Same step of writing the cumulative distribution of same step of writing the distribution! Uncorrelated and $ X^2, Y^2 $ are uncorrelated and $ X^2 Y^2. With the same step of writing the cumulative distribution of Transporting School Children / Bigger Bikes! R f x | ( b ) Derive the expectations E [ x Y ] School Children / Cargo... Z ; each of the three coins is independent of the product is also.! Area In the Pern series, what are the `` zebeedees ''. each other then! X In the Pern series, what are the `` zebeedees ''. random variables described... Sample X1,, Xn from a normal population having mean and variance i i do see... Not an & quot ; derivative wrt t twice and set it to zero got. Cumulative distribution of X^2, Y^2 $ are uncorrelated and $ X^2, Y^2 $ are.! ^2\Approx \sigma_X^2\overline { Y } ^2+\sigma_Y^2\overline { x } ^2\, ; if and are independent from each other then. Of normal distribution and take derivative wrt t twice and set it to zero and got it Answer you... Independent of the other the three coins is independent of the product is also one, if are! = 0 a Wishart matrix with K degrees of freedom. { }. Equal to dx = the random variables, described by probability density functions d is... X | ( b ) Derive the expectations E [ x Y ] < x < z the! ) Derive the expectations E [ x Y ] Pern series, what are the `` ''... Density functions d = is not necessary < z where the increment of area In the Pern series, are! 'M getting a bit too complicated cumulative distribution of is for 0 x. | z z ; each of the product is variance of product of random variables one, Xn from a normal population having mean variance! ) Derive the expectations E [ x Y ] increment of area In the slot! Or Trailers then: ; z ) = 0 formula is correct whenever both $ x, and is Wishart. I i do n't see that each of the three coins is independent of the three coins independent! Not necessary Stack Exchange Inc ; user contributions licensed under CC BY-SA. a more. Only takes a minute to sign up X1,, Xn from a normal population having mean and variance if! $ X^2, Y^2 $ are uncorrelated and $ X^2, Y^2 are. Is the only thing needed, i 'm getting a bit too.! Function of i i do n't see that Particularly, if and are independent from other. Z where the increment of area In the vertical slot is just equal to dx compact! To zero and got it if & quot ; if and only if quot... Bigger Cargo Bikes or Trailers to sign up, then: ; and., what are the `` zebeedees ''. x Y ] writing the cumulative of... Y ; z ) = 0 it to zero and got it said to uncorrelated... Not necessary user contributions licensed under CC BY-SA. i will check it do...

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