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# variance

# Definition

The *variance* of a real-valued random variable $X$ is

$\Var X=\mathbb{E}\bigl[(X-m)^{2}\bigr]\,,\quad m=\mathbb{E}X\,,$ |

provided that both expectations $\mathbb{E}X$ and $\mathbb{E}[(X-m)^{2}]$ exist.

The variance of $X$ is often denoted by $\sigma^{2}(X)$, $\sigma^{2}_{X}$, or simply $\sigma^{2}$. The exponent on $\sigma$ is put there so that the number $\sigma=\sqrt{\sigma^{2}}$ is measured in the same units as the random variable $X$ itself.

The quantity $\sigma=\sqrt{\Var X}$ is called the *standard deviation*
of $X$;
because of the compatibility of the measuring units,
standard deviation is usually the quantity that is quoted
to describe an emprical probability distribution, rather than the variance.

# Usage

It is not always the best measure of dispersion for all random variables, but compared to other measures, such as the absolute mean deviation, $\mathbb{E}[\lvert X-m\rvert]$, the variance is the most tractable analytically.

The variance is closely related to the $\mathbf{L}^{2}$ norm for random variables over a probability space.

# Properties

1. 2. Variance is not a linear function. It scales quadratically, and is not affected by shifts in the mean of the distribution:

$\Var[aX+b]=a^{2}\Var X\,,\quad\text{ for any $a,b\in\mathbb{R}$.}$ 3. A random variable $X$ is constant almost surely if and only if $\Var X=0$.

4. The variance can also be characterized as the minimum of expected squared deviation of a random variable from any point:

$\Var X=\inf_{{a\in\mathbb{R}}}\mathbb{E}[(X-a)^{2}]\,.$ 5. For any two random variables $X$ and $Y$ whose variances exist, the variance of the linear combination $aX+bY$ can be expressed in terms of their covariance:

$\Var[aX+bY]=a^{2}\Var X+b^{2}\Var Y+2ab\Cov[X,Y]\,,$ where $\Cov[X,Y]=\mathbb{E}[(X-\mathbb{E}X)(Y-\mathbb{E}Y)]$, and $a,b\in\mathbb{R}$.

6. For a random variable $X$, with actual observations $x_{1},\ldots,x_{n}$, its variance is often estimated empirically with the

*sample variance*:$\Var X\approx s^{2}=\frac{1}{n-1}\sum_{{i=1}}^{n}(x_{i}-\bar{x})^{2}\,,\quad% \bar{x}=\frac{1}{n}\sum_{{j=1}}^{n}x_{j}\,.$

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## Comments

## hi there i need some assistance

Let X have the cumulative distribution function

F(x) = 1-x^(-Î±) , x >= 1.

a) Find E(X) for those values of X for which E(X) exists.

b) Find Var(X) for those values of X for which Var(X) exists

this question is about probability

thankyou.

## Re: hi there i need some assistance

What is -ÃŽÂ±?