Welcome! Make sure you have access to R. I recommend that you install it!
Download/Install R
Download/Install R studio
Download/Install R markdown
Install R bookdown
Install Tinytex
It maybe possible to do everything for free or a small price over the cloud
Not only can you type run R code, you can produce html and pdf files with math and R code together! Roughly speaking R studio is an easy way to use R, and R markdown is a language easily generates html files with R code and math symbols, allowing the use of basic Latex code; it is how I generates many of the html pages for our module.
Try writing up solutions to these questions in R-Markdown.
Let \(X\) and \(Y\) be independent fair dice. Compute the conditional pmf of \(X\) given that \(X+Y = 7\).
Let \(X\) and \(Y\) be independent random variables, where \(X\) is Bernoulli with parameter \(p \in [0,1]\) and \(Y\) is Poisson with mean \(1\). Let \(Z = X+Y\). What is the probability that \(Z=1\)?
Let \(X = (X_1, \ldots, X_n)\) be a random sample from the Bernoulli family with parameter \(p \in (0,1)\). Consider a hypothesis test with \(H_0: p =1/2\) vs \(H_1: p =1/4\) and the test statistic \(T= \bar{X}\). Let \(n=25\). What should the rejection region be for a test of significance level \(0.05\)?
Let \(\Phi\) be the cdf for the standard normal distribution. Compute the value of the following integral in terms of \(\Phi\). \[\int_{-1} ^{1} e^{-(x-5)^2} dx.\]
Let \(Z\) be a standard normal random variable. Find an explicit deterministic function \(\phi\) such that \(\phi(Z)\) is uniformly distributed in \(\{1,2,3\}.\)
Let \(U\) be uniformly distributed in on a disc of radius \(1\) centered at the origin in \(\mathbb{R}^2\). Write \(U = (X,Y)\). Show explicitly that \(X\) is not independent of \(Y\).
Let \(Z\), \(\epsilon\), and \(\delta\) be independent random variables. Suppose that \(Y = \phi(Z, \epsilon)\) and \(X = \psi(Z, \delta)\), where \(\phi\) and \(\psi\) are deterministic functions. Carefully show that \(X\) is independent of \(Y\), given \(Z\), in the case that all the random variables are discrete.
Let \(U_1, \ldots, U_n\) be an independent discrete sequence of random variables that uniformly distributed in \(\{1,2,3\}\). Consider \(M_n = \max(U_1, \ldots, U_n)\). Prove that \(M_n\) converges to \(3\) as \(n \to \infty\).
Consider a sequence of \(20\) fair coin flips, Estimate the probability that we will see a run of at least four heads. It is not hard to get a lower bound of \(0.27\).
Let \(X\) and \(Y\) be independent random variables that are uniformly distributed in \([0,1]\). Sketch the pdf of \(Z = X+Y\).
Let \(X_1, \ldots, X_n\) be independent Poisson random variables with mean \(\lambda\). Let \(T = X_1 + \cdots + X_n\). Let \(Y = \mathbf{1}[X_1=1]\) be the Bernoulli random variable that takes the value \(1\) if and only if \(X_1=1\). Show that \(Z=\mathbb{E}(Y |T)\) converges to \(\mathbb{P}(X_1=1)\) as \(n \to \infty\).
Let \(\epsilon_1\) and \(\epsilon_2\) be independent standard normals. Consider the system of equations \[\begin{eqnarray*} X &=& \tfrac{1}{2} Y + \epsilon_1 \\ Y &=& \tfrac{1}{2} X + \epsilon_2. \end{eqnarray*}\] Does the system have a solution, \((X,Y)\); what would the distribution be?