Chapter 1

Let \(f(x)=1 / x^{2}, 1

Let the space of the random variable \(X\) be \(\mathcal{D}=\\{x: 0

Let \(X\) have the pdf \(f(x)=3 x^{2}, 0

Let \(X\) have a pmf \(p(x)=\frac{1}{3}, x=1,2,3\), zero elsewhere. Find the pmf of \(Y=2 X+1\)

Let \(X\) be a positive random variable; i.e., \(P(X \leq 0)=0\). Argue that (a) \(E(1 / X) \geq 1 / E(X)\) (b) \(E[-\log X] \geq-\log [E(X)]\) (c) \(E[\log (1 / X)] \geq \log [1 / E(X)]\) (d) \(E\left[X^{3}\right] \geq[E(X)]^{3}\).

In a certain factory, machines I, II, and III are all producing springs of the same length. Machines I, II, and III produce \(1 \%, 4 \%\), and \(2 \%\) defective springs, respectively. Of the total production of springs in the factory, Machine I produces \(30 \%\), Machine II produces \(25 \%\), and Machine III produces \(45 \%\). (a) If one spring is selected at random from the total springs produced in a given day, determine the probability that it is defective. (b) Given that the selected spring is defective, find the conditional probability that it was produced by Machine II.

A mode of the distribution of a random variable \(X\) is a value of \(x\) that maximizes the pdf or pmf. If there is only one such \(x\), it is called the mode of the distribution. Find the mode of each of the following distributions: (a) \(p(x)=\left(\frac{1}{2}\right)^{x}, x=1,2,3, \ldots\), zero elsewhere. (b) \(f(x)=12 x^{2}(1-x), 0

Let \(X\) be a random variable such that \(E\left[(X-b)^{2}\right]\) exists for all real \(b\). Show that \(E\left[(X-b)^{2}\right]\) is a minimum when \(b=E(X)\).

Let \(X\) have the \(\operatorname{pmf} p(x)=\left(\frac{1}{2}\right)^{x}, x=1,2,3, \ldots\), zero elsewhere. Find the pmf of \(Y=X^{3}\).

Suppose the random variable \(X\) has the cdf $$ F(x)=\left\\{\begin{array}{ll} 0 & x<-1 \\ \frac{x+2}{4} & -1 \leq x<1 \\ 1 & 1 \leq x \end{array}\right. $$ Write an \(\mathrm{R}\) function to sketch the graph of \(F(x)\). Use your graph to obtain the probabilities: (a) \(P\left(-\frac{1}{2}