Chapter 1

Let \(0

For every two-dimensional set \(C\) contained in \(R^{2}\) for which the integral exists, let \(Q(C)=\iint_{C}\left(x^{2}+y^{2}\right) d x d y .\) If \(C_{1}=\\{(x, y):-1 \leq x \leq 1,-1 \leq y \leq 1\\}\), \(C_{2}=\\{(x, y):-1 \leq x=y \leq 1\\}\), and \(C_{3}=\left\\{(x, y): x^{2}+y^{2} \leq 1\right\\}\), find \(Q\left(C_{1}\right), Q\left(C_{2}\right)\) and \(Q\left(C_{3}\right)\).

Let \(X\) denote a random variable for which \(E\left[(X-a)^{2}\right]\) exists. Give an example of a distribution of a discrete type such that this expectation is zero. Such a distribution is called a degenerate distribution.

Two distinct integers are chosen at random and without replacement from the first six positive integers. Compute the expected value of the absolute value of the difference of these two numbers.

In an office there are two boxes of thumb drives: Box \(A_{1}\) contains seven 100 GB drives and three 500 GB drives, and box \(A_{2}\) contains two 100 GB drives and eight 500 GB drives. A person is handed a box at random with prior probabilities \(P\left(A_{1}\right)=\frac{2}{3}\) and \(P\left(A_{2}\right)=\frac{1}{3}\), possibly due to the boxes' respective locations. A drive is then selected at random and the event \(B\) occurs if it is a \(500 \mathrm{~GB}\) drive. Using an equally likely assumption for each drive in the selected box, compute \(P\left(A_{1} \mid B\right)\) and \(P\left(A_{2} \mid B\right)\)

A bowl contains 16 chips, of which 6 are red, 7 are white, and 3 are blue. If four chips are taken at random and without replacement, find the probability that: (a) each of the four chips is red; (b) none of the four chips is red; (c) there is at least one chip of each color.

Let \(X\) have a Cauchy distribution which has the pdf $$ f(x)=\frac{1}{\pi} \frac{1}{x^{2}+1}, \quad-\infty

For each of the following cdfs \(F(x)\), find the pdf \(f(x)[\mathrm{pmf}\) in part \((\mathbf{d})]\), the first quartile, and the \(0.60\) quantile. Also, sketch the graphs of \(f(x)\) and \(F(x)\). May use \(\mathrm{R}\) to obtain the graphs. For Part(a) the code is provided. (a) \(F(x)=\frac{1}{2}+\frac{1}{\pi} \tan ^{-1}(x),-\infty

Let \(\mathcal{C}\) denote the set of points that are interior to, or on the boundary of, a square with opposite vertices at the points \((0,0)\) and \((1,1)\). Let \(Q(C)=\iint_{C} d y d x\). (a) If \(C \subset \mathcal{C}\) is the set \(\\{(x, y): 0

Let \(X\) denote a random variable such that \(K(t)=E\left(t^{X}\right)\) exists for all real values of \(t\) in a certain open interval that includes the point \(t=1 .\) Show that \(K^{(m)}(1)\) is equal to the \(m\) th factorial moment \(E[X(X-1) \cdots(X-m+1)]\).