Consider a continuous (latent) variable $Y^*$ given as follows: $Y^*=\beta'x+\epsilon$ and calculate different probabilities...

Question

Consider a continuous (latent) variable $Y^*$ given as follows: $Y^*=\beta'x+\epsilon$,
where $\beta'x=\beta_0+\beta_1x_1+\beta_2x_2+...+\beta_px_p$ and $\epsilon$ ~ $N(0,1)$ is independent of $x$. Define further $Y$ as the indicator: $Y= 1$ if $ Y^*>0$ , i.e. $-\epsilon<\beta'x$, or $Y=0$ otherwise.

a) Show that for all real $u$,
$P(-\epsilon ≤ u)=P(\epsilon≤u)$.

b)Show that
$P(Y=1|x)=\phi(\beta'x)$
where $\phi()$ is the distribution function of $N(0,1)$.

Any help would be much appreciated.

Answer 1

1. Note that $\epsilon$ is standard normal random variable, thus its symmetric around $0$ thus the result is trivial. Formally, $$ \mathbb{P}(-\epsilon \le u) = \mathbb{P}(\epsilon \ge - u) = 1 - \Phi(-u) = 1-(1-\Phi(u))=\Phi(u)=\mathbb{P}(\epsilon \le u). $$

2. Just use the resulf from (1). $$ \mathbb{P}(Y=1|X=x) = \mathbb{P}(-\epsilon \le x'\beta) = \Phi(x'\beta) $$