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 \(1\).
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 \(0.27\).
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 \(\mathbb{E} T\).