PROBABILITIES Course

Variance

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The variance is the square deviation around the expected value.

\[ \begin{align}\begin{aligned}V(X) = \mathbb{E}[(X - \mathbb{E}[X])^2]\\V(X) = \mathbb{E}[X^2] - \mathbb{E}[X]^2\end{aligned}\end{align} \]

We got the second formula with Huygens theorem.


Properties

  • $V(X) = \sigma^2$ with sigma the standard deviation/écart-type
  • $V(\lambda X^2) = \lambda^2 V(X)$
  • $V(c) = 0$ (or if $V(X)=0$ then X is a constant)
  • $V(a + \lambda X^2) = \lambda^2 V(X)$
  • $V(X + Y) = V(X) + V(Y) - 2cov(XY)$
  • $V(X)$ is also called the second central moment (moment (centré) de second ordre)

Chebyshev's inequality

@ \mathbb{P}(|X - \mathbb{E}[X]| \ge a) \le \frac{V[X]}{a^2} @


Covariance/co-variance (cov)

It's used to evaluate the conjoint variance of two random variables.

\[ \begin{align}\begin{aligned}cov(x,y) = \mathbb{E}[ ( X - \mathbb{E}[X]) (Y - \mathbb{E}[Y]) ]\\cov(x,y) = \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y]\end{aligned}\end{align} \]

Properties

  • $cov(X,X) = V(X)$
  • $cov(X,Y) = cov(Y,X)$
  • $cov(\lambda * X,Y) = \lambda *cov(Y,X)$
  • $cov(\lambda * X) = \lambda^2 *cov(X)$
  • $cov(A+B,C) = cov(A,C) + cov(B,C)$
  • if $X \perp Y$ then $cov(XY) = 0$
  • $\mathbb{P}(X, Y) = \frac{cov(X,Y)}{\sqrt{V(X)*V(Y)}}$