Suppose you have at hand a pseudo-random number generator that can simulate an i.i.d. sequence of uniform $U[0,1]$ distributed random variables $U_{1}^{*}, \ldots, U_{N}^{*}$ for any $N \in \mathbb{N}$. Construct an algorithm to simulate an i.i.d. sequence $X_{1}^{*}, \ldots, X_{N}^{*}$ of standard normal $N(0,1)$ random variables. [Should your algorithm depend on the inverse of any cumulative probability distribution function, you are required to provide an explicit expression for this inverse function.]

Suppose as a matter of urgency you need to approximately evaluate the integral

$$I=\frac{1}{\sqrt{2 \pi}} \int_{\mathbb{R}} \frac{1}{(\pi+|x|)^{1 / 4}} e^{-x^{2} / 2} d x$$

Find an approximation $I_{N}$ of this integral that requires $N$ simulation steps from your pseudo-random number generator, and which has stochastic accuracy

$$\operatorname{Pr}\left(\left|I_{N}-I\right|>N^{-1 / 4}\right) \leqslant N^{-1 / 2}$$

where Pr denotes the joint law of the simulated random variables. Justify your answer.