Chapter 7

Let \(T\) be the outcome of a roll with a fair die. a. Describe the probability distribution of \(T\), that is, list the outcomes and the corresponding probabilities. b. Determine \(\mathrm{E}[T]\) and \(\operatorname{Var}(T)\).

The probability distribution of a discrete random variable \(X\) is given by $$ \mathrm{P}(X=-1)=\frac{1}{5}, \quad \mathrm{P}(X=0)=\frac{2}{5}, \quad \mathrm{P}(X=1)=\frac{2}{5} . $$ a. Compute \(\mathrm{E}[X]\). b. Give the probability distribution of \(Y=X^{2}\) and compute \(\mathrm{E}[Y]\) using the distribution of \(Y\). c. Determine \(\mathrm{E}\left[X^{2}\right]\) using the change-of-variable formula. Check your answer against the answer in \(\mathbf{b}\). d. Determine \(\operatorname{Var}(X)\).

For a certain random variable \(X\) it is known that \(\mathrm{E}[X]=2\), \(\operatorname{Var}(X)=3\). What is \(\mathrm{E}\left[X^{2}\right]\) ?

Let \(X\) be a random variable with \(\mathrm{E}[X]=2\), \(\operatorname{Var}(X)=4\). Compute the expectation and variance of \(3-2 X\).

Determine the expectation and variance of the \(\operatorname{Ber}(p)\) distribution.

The random variable \(Z\) has probability density function \(f(z)=3 z^{2} / 19\) for \(2 \leq z \leq 3\) and \(f(z)=0\) elsewhere. Determine \(\mathrm{E}[Z]\). Before you do the calculation: will the answer lie closer to 2 than to 3 or the other way around?

Given is a random variable \(X\) with probability density function \(f\) given by \(f(x)=0\) for \(x<0\), and for \(x>1\), and \(f(x)=4 x-4 x^{3}\) for \(0 \leq x \leq 1\). Determine the expectation and variance of the random variable \(2 X+3\).

Given is a continuous random variable \(X\) whose distribution function \(F\) satisfies \(F(x)=0\) for \(x<0, F(x)=1\) for \(x>1\), and \(F(x)=x(2-x)\) for \(0 \leq x \leq 1\). Determine \(\mathrm{E}[X]\).

Let \(U\) be a random variable with a \(U(\alpha, \beta)\) distribution. a. Determine the expectation of \(U\). b. Determine the variance of \(U\).

Let \(X\) have an exponential distribution with parameter \(\lambda\). a. Determine \(\mathrm{E}[X]\) and \(\mathrm{E}\left[X^{2}\right]\) using partial integration. b. Determine \(\operatorname{Var}(X)\).

In this exercise we take a look at the mean of a Pareto distribution. a. Determine the expectation of a \(\operatorname{Par}(2)\) distribution. b. Determine the expectation of a \(\operatorname{Par}\left(\frac{1}{2}\right)\) distribution. c. Let \(X\) have a \(\operatorname{Par}(\alpha)\) distribution. Show that \(\mathrm{E}[X]=\alpha /(\alpha-1)\) if \(\alpha>1\).

For which \(\alpha\) is the variance of a \(\operatorname{Par}(\alpha)\) distribution finite? Compute the variance for these \(\alpha\).

Remember that we found on page 95 that the expected area of a building was \(33 \frac{1}{3} \mathrm{~m}^{2}\), whereas the square of the expected width was only \(25 \mathrm{~m}^{2}\). This phenomenon is more general: show that for any random variable \(X\) one has \(\mathrm{E}\left[X^{2}\right] \geq(\mathrm{E}[X])^{2} .\)

Suppose we choose arbitrarily a point from the square with corners at \((2,1),(3,1),(2,2)\), and \((3,2)\). The random variable \(A\) is the area of the triangle with its corners at \((2,1),(3,1)\), and the chosen point. (See also Exercise \(5.9\) and Figure 7.5.) Compute \(\mathrm{E}[A]\).

The probability density function \(f\) of the random variable \(X\) used in Figure \(7.2\) is given by \(f(x)=0\) outside \((0,1)\) and \(f(x)=-4 x \ln (x)\) for \(0

Let \(U\) be a discrete random variable taking the values \(a_{1}, \ldots, a_{r}\) with probabilities \(p_{1}, \ldots, p_{r} .\) a. Suppose all \(a_{i} \geq 0\), but that \(\mathrm{E}[U]=0\). Show then $$ a_{1}=a_{2}=\cdots=a_{r}=0 . $$ In other words; \(\mathrm{P}(U=0)=1\).