The intuition can be seen in the simple case var(x+y): if x and y are positively correlated, both will tend to be large/small together, increasing total variation. If they are negatively correlated, they will tend to cancel each other, decreasing total variation. $\endgroup$

In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value.

2018年10月9日 · $var (x-y) = $$e[(x-y)^2] - e[x-y]^2\\ e[x^2 - 2xy + y^2] - e[x-y]^2$ Linearity of expectation: $E[X^2 - 2XY + Y^2] = E[X^2] + E[Y^2] - 2E[XY]$ and $E[X-Y] = E[X] - E[Y]$

2014年4月29日 · Write out the variance as much as you can, then look for quantities with known values. We start from. Var[X − Y] =E[(X − Y)2] − (E[X − Y])2. Var [X − Y] = E [(X − Y) 2] − (E [X − Y]) 2. Now, we can multiply these out and use linearity of the expectation to get: Var[X − Y] =E[X2] − 2E[XY] +E[Y2] − (E[X])2 + 2E[X]E[Y] − (E[Y])2.

E(X+Y) = E(X) + E(Y) = 7\\ Var(X+Y) = Var(X) + Var(Y) = 7. X-Y 表示两个骰子面值之差. X，Y 取值范围在 1-6，而 X-Y的取值范围在 -5 - 5，数据范围变大，变异性自然变大。 X-Y的概率分布

4. Variance of convolution of random variables: Given two random variables X and Y, variance of their sum is: Var(X +Y)=Var(X)+Var(Y)+2Cov(X;Y) (Theorem 5). If, however, X and Y are independent, then we have Var(X +Y)=Var(X)+Var(Y) since the covariance of independent random variables is always zero.

2017年4月21日 · Let $X,Y$ be random variables with variances $\sigma^{2}_{x}$ and $\sigma^{2}_{y}$, respectively. It is a fact that ${\rm var}(Z) = {\rm cov}(Z,Z)$ for any random variable $Z$. This can be checked using the definition of covariance and variance. So, the variance of $X-Y$ is $$ {\rm cov}(X-Y,X-Y) = {\rm cov}(X,X)+{\rm cov}(Y,Y)-2\cdot{\rm cov}(X ...

For example, suppose a random variable X can take values x 1;x 2;::: and that X is independent of another random variable Y. Consider the expected value of a product g(X)h(Y), for any functions gand h. Calculate by conditioning on the possible values taken by X: Eg(X)h(Y) = X i PfX= x igE(g(X)h(Y) jX= x i): Given that X= x i, we know that g(X ...

V (X + Y) = V (X) + V ( Y) + 2Cov(X, Y) where Cov(X, Y) is the covariance between X and Y, Cov(X, Y) = E( (X− E(X)) ( Y − E( Y)) ). If X and Y are independent (or merely uncorrelated) then Cov(X, Y) = 0. This additive rule for variances extends to three or more random variables; e.g., V (X + Y + Z) = V (X) + V ( Y) + V (Z) +2Cov(X, Y ...

variables X and Y. Covariance can be either positive or negative. (Variance is always positive.) Definition: Let X and Y be any random variables. The covariance between X and Y is given by cov(X,Y) = E n (X − µ X)(Y − µ Y) o = E(XY)− E(X)E(Y), whereµ X = E(X), µ Y = E(Y). 1. cov(X,Y) will be positiveif large values of X tend to occur ...