# B Probability Reminders

## B.1 Independence

Definition B.1 Two random variables $$X$$ and $$Y$$ are (statistically) independent, if $$P(X\in A, Y\in B) = P(X\in A) P(Y\in B)$$ for all sets $$A$$ and $$B$$.

We list some properties of independent random variables:

• If $$X$$ and $$Y$$ are independent, and if $$f$$ and $$g$$ are functions, then $$f(X)$$ and $$g(Y)$$ are also independent.

## B.2 The Chi-Squared Distribution

Definition B.2 Let $$X_1, \ldots, X_\nu \sim \mathcal{N}(0, 1)$$ be i.i.d. Then the distribution of $$\sum_{i=1}^\nu X_i^2$$ is called the $$\chi^2$$-distribution with $$\nu$$ degrees of freedom. The distribution is denoted by $$\chi^2(\nu)$$.

Some important properties of the $$\chi^2$$-distribution are:

• $$\chi^2$$-distributed random variables are always positive.

• If $$Y\sim \chi^2(\nu)$$, then $$\mathbb{E}(Y) = \nu$$ and $$\mathop{\mathrm{Var}}(Y) = 2\nu$$.

• The R command pchisq(|$$x$$,$$\nu$$) gives the value $$\Phi_\nu(x)$$ of the CDF of the $$\chi^2(\nu)$$-distribution.

• The R command qchisq($$\alpha$$,$$\nu$$) can be used to obtain the $$\alpha$$-quantile of the $$\chi^2(\nu)$$-distribution.

• More properties can be found on Wikipedia.

## B.3 The t-distribution

Definition B.3 Let $$Z \sim \mathcal{N}(0,1)$$ and $$Y \sim \chi^2(\nu)$$ be independent. Then the distribution of $$$T = \frac{\,Z\,}{\,\sqrt{Y / \nu}\,} \tag{B.1}$$$ is called the $$t$$-distribution with $$\nu$$ degrees of freedom. This distribution is denoted by $$t(\nu)$$.

Some important properties of the $$t$$-distribution are:

• The $$t$$-distribution is symmetric: if $$T \sim t(\nu)$$, then $$-T \sim t(\nu)$$

• If $$T\sim t(\nu)$$, then $$\mathbb{E}(T) = 0$$.

• The R command pt(|$$x$$,$$\nu$$) gives the value $$\Phi_\nu(x)$$ of the CDF of the $$t(\nu)$$-distribution.

• The R command qt($$\alpha$$,$$\nu$$) can be used to obtain the $$\alpha$$-quantile of the $$t(\nu)$$-distribution.

• More properties can be found on Wikipedia.