Q:

Contaminated normals, Multivariate normal distributions and PCA

While studying the above mentioned topics, i got a little confused in reading two things. I have two questions. First, in:
Scanned text 1
how exactly P[W <= w] unfolds as seen in the red rectangle?
And secondly:
Scanned text 2
a_i? I suppose this could be a typo. In case not, please let me know.
Thank you.

A:

Regarding $1$:
This is the principle of exhaustion. either $I_\epsilon = 1$ or $I_\epsilon = 0$ but not both and not neither. Thus
$$P(\Phi) = P(\Phi|I_\epsilon = 1) + P(\Phi | I_\epsilon = 0)$$
Regarding $2$:
This is no typo, it's the inner product
$$a' v_i = a^T v_i = \langle a, v_i \rangle$$
Just some odd notation ($\cdot'$ for the transposed row-vector)
The inner product of two vectors $u,v \in \mathbb R^n$ is defined by
$$\langle u,v \rangle := \sum_{i=1}^n u_i v_i$$
note the different meaning of $v_i$: here $v_i$ denotes the $i$-th component of the vector $v$ and above $v_i$ is itself a vector

