这是一份 Imperial帝国理工大学 MATH97228作业代写的成功案例
one can show that
$$
\mu_{X}[a, b]=b-a, \quad 0 \leq a \leq b \leq 1
$$
in other words, the distribution measure of $X$ is uniform on $[0,1]$.
(Standand normal random variable). Let
$$
\varphi(x)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{x^{2}}{2}}
$$
be the standard normal density, and define the cumulative normal distribution function
$$
N(x)=\int_{-\infty}^{x} \varphi(\xi) d \xi
$$
MATH97228 COURSE NOTES ：
Let $X$ be a random variable defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. We would like to compute an “average value” of $X$, where we take the probabilities into account when doing the averaging. If $\Omega$ is finite, we simply define this average value by
$$
\mathbb{E} X=\sum_{\omega \in \Omega} X(\omega) \mathbb{P}(\omega)
$$
If $\Omega$ is countably infinite, its elements can be listed in a sequence $\omega_{1}, \omega_{2}, \omega_{3}, \ldots$, and we can define $\mathbb{E X}$ as an infinite sum:
$$
\mathbb{E} X=\sum^{\infty} X\left(\omega_{k}\right) \mathbb{P}\left(\omega_{k}\right)
$$