PROBABILITIES Course

Expected value and variance

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The expected value $\mathbb{E}(X)$ is defined by:

@ \mathbb{E}[X] = \int_{-\infty}^{+\infty} xf_X(x)dx @

  • if the density function is not integrable, then the expected value do not exist
  • if $\mathbb{E}(X) = 0$ then $X$ is centered
  • if $\mathbb{E}(X)$ is finite, then $X$ may be centered
  • $\mathbb{E}[X * Y] = \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} x * y * f_{XY}(x , y)\ dx dy$

Variance

The variance $V(X)$ is defined by

@ V[X] = \int_{-\infty}^{+\infty} (x-\mathbb{E}[X])^2\ f_X(x)\ dx @