```{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 $\mathbb{E} T$.





# Version: 5 October 2020

*  The R Markdown source file is available at [this link](https://tsoo-math.github.io/ucl/QHW1.Rmd)

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