这是一份ed.ac爱丁堡格大学MATH11132作业代写的成功案

fact take place, estimate the probabilities (which will clearly sum to 1) for each outcome from no injury up to death. In an obvious notation we can now obtain the valuc of risk as:
$$
R=\sum_{n=1}^{2} p(n) \sum_{3=1}^{6} p(n, s) w(s)
$$
where $p(1)$ and $p(2)$ are the respective probabilities of a bad fall and avalanche is, for example, the probability that an avalanche, if it occurs, will lead to death. The generalisation to a larger number of adverse occurrences is obvious. This measure of risk $R$ can be called, for obvious reasons, the “equivalent probability of death”, with the minimum value of 0 corresponding to no possibility of injury or death and the maximum value of 1 corresponding to imminent death with certainty.

MATH11132 COURSE NOTES ：

A Gaussian of mean $m$ and root mean square $\sigma$ is defined as:
$$
P_{G}(x) \equiv \frac{1}{\sqrt{2 \pi \sigma^{2}}} \exp \left(-\frac{(x-m)^{2}}{2 \sigma^{2}}\right) \text {. }
$$
The median and most probable value are in this case equal to $m$, while the MAD (or any other definition of the width) is proportional to the RMS (for example, $E_{a h m}=\sigma \sqrt{2 / \pi}$ ). For $m=0$, all the odd moments are zero while the even moments are given by $m_{2 n}=(2 n-1)(2 n-3) \ldots \sigma^{2 n}=$ $(2 n-1) ! ! \sigma^{2 n}$.

All the cumulants of order greater than two are zero for a Ganssian. This can be realised by examining its characteristic function:
$$
\hat{P}_{C}(z)=\exp \left(-\frac{\sigma^{2} z^{2}}{2}+i m z\right)
$$