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I know that the variance of a linear combination of correlated random variables can be generalized (as in Variance of linear combinations of correlated random variables). … My question has to do with the variance of an additional linear combination of two or more such linear combinations. …

$ a^Tx:=\sum_{i=1}^na_ix_i $ $ \sum a_i^2=1 $ how do I show that the linear combination of 1 and 2 has maximal variance when a is an eigenvector of $\Sigma$ with maximal eigenvalue? …

combination of these bivariate normal distributions with weights $c = [c_1,...,c_i]$ where $\sum c_i = 1$ & $c_i >0$ Obviously, the linear combination of $\mu_{mixture} = [\sum c_ix_i,\sum c_iz_i]$ However … , I am not sure about the linear combination of the variance-covariance matrix. …

I have rewritten the above equation at $Y=-X$, inserting $-X$ for $Y$: $Z=b_0+(b_1-b_2+b_9V+b_{10}W-b_{11}V)X+(b_5-b_7)X^2+e$ Thus, the linear combination of coefficients for the slope along $Y=-X$ … 2b_{10}+2Vcovb_2b_{11}+2VWcovb_9b_{10}-2V^2covb_9b_{11}-2VWcovb_{10}b_{11}$ However, somehow I end up with a negative variance when inserting values. …

$||t||_2 = 1$, suppose we are interested in maximizing the variance of a linear combination of the columns of $\mathbf{A}$, i.e. $\mathbf{A}t$. …

The context is linear regressions and calculating the variance of the prediction. I understand in general, in linear combination of variance, Cs are weights but what are the weights here? …

team productivity, X the productivity of most experienced programmers and Y the productivity of less experienced programmers: $X \sim N(50, 15^2)$ $Y \sim N(30, 10^2)$ $P = 10X + 20Y$ Since P is a linear … combination of Normal distributions: $E[P] = 10E[X] + 20E[Y] = 10(50) + 20(30) = 1100$ $Var(P) = 10^2Var[X] + 20^2Var[Y] = 10^2(15^2) + 20^2(10^2) = 62500$ (...) …

If I understand correctly, unrotated PC1 tells me what linear combination of these variables describes/explains the most variance in the data and PC2 tells me what linear combination of these variables … Let's say I choose some linear combination of these variables -- e.g. $A+2B+5C$, could I work out how much variance in the data this describes? …

Because I clearly obtain less variance than intuitively expected when combining distributions. …

Or alternatively, what amount of variance of $Y$ is unique, that is, what amount of variance is not explained by $X_1$ and $X_2$? … Edit: Simplified my problem too much, and jbowman correctly pointed out all the variance is explained if X1 and X2 are linear combinations of just two random variables, so added a third random variable …

Let $ \{X_t\}$ ~ AR(1): $$ X_t=2.62-0.84X_{t-1}+\epsilon_t, \ \ \ \epsilon_t\sim WN(0,2.27)$$ Compute the variance of $$ \overline{X}= \frac{1}{3}\sum_{t=1}^{3} X_t $$ The solution is: Var($\overline …

Suppose now that $y$ is exactly replicated by $\beta x$ ,i.e. each row of $y$ is a linear combination of the corresponding row in $x$ (for example assume $y=3*Ix$ where $I$ is the unit matrix). … However, I am doing nothing wrong, as I am finding the true coefficients of the data generating process because, as said, y is a linear combination of x. …

Let $X_1, X_2,...,X_{2n}$ be random variables such that $V(X_i)=4, i=1,2,...,2n$ and $Cov(X_i, X_j)=3, 1\leq i\neq j\leq 2n$. Then find $V(X_1-X_2+X_3-X_4+...+X_{2n-1}-X_{2n})$. I know that $V( …

Let $A$ and $B$ be two constant matrices and let $x$ and $ y$ be two random vectors, what is the general formula for $Var(Ax+By)$? I know the formula for when $x$ and $y$ are scalar random variables a …

Suppose I have an index $X_t$ over time, which is a linear combination of $N$ other time-series $x_{i,t}$. So $X_t= \sum_i^n w_{i}x_{i,t}$,. … For variances this is clear, there we can decompose the variance of an index in terms of the variances of the underlying variables and their corresponding co-variances. …