I'm trying to solve the following problem:
Let $X_n \sim \mathcal{N}(0,\frac{1}{n})$, and let $Y_n$ be the variable defined by: $$Y_n(\omega)=\int_{-1}^1 | X_n(\omega)-t |\,dt $$
Let $F_{Y_n}$ denote the cumulative distribution function of $Y_n$, and by $\Phi$ the cumulative distribution function of the standard normal distribution.
Question: For each $y \in \mathbb{R}$ write $F_{Y_n}(y)$ in terms of $\Phi(y)$ and $n$.
Here is the sketch of my uncompleted solution :
First we can note that $Y_n = X_n^2+1$. Thus, for $y>1$, $$F_{Y_n}(y) = P(X_n^2+1 < y) = F_{X_n}(\sqrt{y-1})-F_{X_n}(-\sqrt{y-1}).$$
Now I have to write $F_{X_n}(\sqrt{y-1})-F_{X_n}(-\sqrt{y-1})$ in terms of $\Phi(y)$ and $n$. But I don't know how to do it.
Maybe we can use the fact that $\Phi(x)\; \sim_0\;0.5+\frac{1}{\sqrt{2\pi}}\cdot e^{-x^2/2}\left[x+\frac{x^3}{3}+\frac{x^5}{3\cdot 5}+...+\frac{x^{2n+1}}{3\cdot 5\cdot7\cdot ...\cdot (2n+1)}\right]$