# Integration

Approximate the integral \[ \int_0 ^{\infty} \sin(x) x^2 e^{-x} dx\] by appealing the law of large numbers and using R. Hint: Consider an i.i.d. sequence of exponential random variables all with rate \(1\).

# Grouping coins

Again, consider a sequence of 20 fair coin flips, as discussed in our first live session. using R, estimate the probability that we will see a run of at least four heads. Hopefully, we get a number bigger than \(0.27\).

# Pen and paper?

Let \((U_i)_{i \in \mathbb{Z} ^+}\) be a sequence of independent random variables that are uniformly distributed in \([0,1]\). Let \[S_n = X_1 + \cdots + X_n.\] Let \[T = \inf\{n \geq 1: S_n >1\}\] so that \(T\) is the first time the sum is greater than \(1\). Use R or pen and paper to compute \(\mathbb{E} T\).

# Python

Now do the same exercises in Python.

### Endnotes

- Version 01 October 2021
- The R Markdown source file is usually available by replacing
*html* with *Rmd*; you can use this link