{r setup, include=FALSE} knitr::opts_chunk$set(echo = TRUE)  # 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 r format(Sys.time(), '%d %B %Y') * The R Markdown source file is usually available by replacing *html* with *Rmd*; you can use [this link](https://tsoo-math.github.io/ucl2/2021-HW-week6.Rmd)