Summarising Continuous RVs

Summarising Continuous Variables

We extend our summary tools to continuous space using integrals.

$E[X] = \int_{-\infty}^{\infty} x f(x) dx$

Properties like linearity and additivity of variance (for independent variables) remain true.

The M.G.F. is defined as $M(t) = E[e^{tX}]$ .

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Moments: $E[X^k]$ can be found by taking the $k$ -th derivative of $M(t)$ at $t=0$ .
Uniqueness: The M.G.F. uniquely determines the distribution. If two variables have the same M.G.F., they have the same distribution.

For Bivariate Normal variables, zero correlation ( $\rho=0$ ) implies independence. This is a unique property of the normal distribution!