Parameter Identification in a (Simple) Mixture Model I have a very mundane clarification question from some old lecture notes. The notes:

Consider the model $$Y_i=(1+D_i)\varepsilon_i$$ where $(D_i,\varepsilon_i)\overset{iid}{\sim}Bernoulli(p)\times N(0,1),p\in(0,1)$, and we observe $Y_i$. The parameter $p$ is point identified.


By the law of total probability $$f_Y(y)=\phi(y)(1-p)+\frac{1}{2}\phi\left(\frac{y}{2}\right)p$$ and solving for $p$, we have...

Where does the $\frac{1}{2}$ factor in front of the second $\phi(\cdot)$ come from? My attempts start with $f_Y(y)=\phi(y)(1-p)+\phi\left(\frac{y}{2}\right)p$, which leads to a different value for $p$.

From the law of total probability we have
$$
P(Y\leq y) = P(Y\leq y \mid D = 0) (1-p) + P(Y\leq y \mid D = 1) p \\
= \Phi(y)(1-p) + \Phi(y/2)p
$$
so
$$
f_Y(y) = \varphi(y)(1-p) + \frac 12 \varphi(y/2)p
$$
by the chain rule.
Regarding your mistake, the law of total probability is about measuring a set by splitting it over a bunch of other sets, so it's at the level of probability measures. Since CDFs are bijective with probability measures on $(\mathbb R, \mathbb B)$ we can just start with $P(Y\leq y)$ rather than $P(Y\in A)$ for a Borel $A$. Only after we have the law of $Y$ confirmed do we get its Lebesgue density, i.e. we differentiate to get $f_Y$.
