variance of product of random variables

( Scaling $Y\cdot \operatorname{var}(X)$ respectively. If X(1), X(2), , X(n) are independent random variables, not necessarily with the same distribution, what is the variance of Z = X(1) X(2) X(n)? ) Variance of product of two random variables ($f(X, Y) = XY$). x ( y You get the same formula in both cases. 1 ) {\displaystyle z} ( So far we have only considered discrete random variables, which avoids a lot of nasty technical issues. &= E[(X_1\cdots X_n)^2]-\left(E[X_1\cdots X_n]\right)^2\\ x , z =\sigma^2\mathbb E[z^2+2\frac \mu\sigma z+\frac {\mu^2}{\sigma^2}]\\ x f The sum of $n$ independent normal random variables. The distribution of the product of correlated non-central normal samples was derived by Cui et al. ) ) 2 ( Find the PDF of V = XY. is a function of Y. }, The variable and let thus. Var(rh)=\mathbb E(r^2h^2)=\mathbb E(r^2)\mathbb E(h^2) =Var(r)Var(h)=\sigma^4 {\displaystyle \theta } assumption, we have that {\displaystyle \theta } {\displaystyle y} What non-academic job options are there for a PhD in algebraic topology? ( {\displaystyle Z=X_{1}X_{2}} n 0 How to tell if my LLC's registered agent has resigned? Thanks for the answer, but as Wang points out, it seems to be broken at the $Var(h_1,r_1) = 0$, and the variance equals 0 which does not make sense. | ) 0 y {\displaystyle (1-it)^{-1}} . = -increment, namely $$. h + f Residual Plots pattern and interpretation? {\displaystyle u=\ln(x)} Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. z Does the LM317 voltage regulator have a minimum current output of 1.5 A? $$ 2 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. f Many of these distributions are described in Melvin D. Springer's book from 1979 The Algebra of Random Variables. i + x y If the first product term above is multiplied out, one of the E x Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators . n rev2023.1.18.43176. EX. {\displaystyle z} Similarly, the variance of the sum or difference of a set of independent random variables is simply the sum of the variances of the independent random variables in the set. K \end{align}$$. X Due to independence of $X$ and $Y$ and of $X^2$ and $Y^2$ we have. | ( I don't see that. ) {\displaystyle f_{Z_{3}}(z)={\frac {1}{2}}\log ^{2}(z),\;\;0

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