Chapter 8

Often one is interested in the distribution of the deviation of a random variable \(X\) from its mean \(\mu=\mathrm{E}[X]\). Let \(X\) take the values \(80,90,100,110\), and 120, all with probability \(0.2 ;\) then \(\mathrm{E}[X]=\mu=100\). Determine the distribution of \(Y=|X-\mu|\). That is, specify the values \(Y\) can take and give the corresponding probabilities.

Suppose \(X\) has a uniform distribution over the points \(\\{1,2,3,4,5,6\\}\) and that \(g(x)=\sin \left(\frac{\pi}{2} x\right)\). a. Determine the distribution of \(Y=g(X)=\sin \left(\frac{\pi}{2} X\right)\), that is, specify the values \(Y\) can take and give the corresponding probabilities. b. Let \(Z=\cos \left(\frac{\pi}{2} X\right)\). Determine the distribution of \(Z\). c. Determine the distribution of \(W=Y^{2}+Z^{2}\). Warning: in this example there is a very special dependency between \(Y\) and \(Z\), and in general it is much harder to determine the distribution of a random variable that is a function of two other random variables. This is the subject of Chapter 11 .

The continuous random variable \(U\) is uniformly distributed over \([0,1]\). a. Determine the distribution function of \(V=2 U+7\). What kind of distribution does \(V\) have? b. Determine the distribution function of \(V=r U+s\) for all real numbers \(r>0\) and \(s\). See Exercise \(8.9\) for what happens for negative \(r\).

Transforming exponential distributions. a. Let \(X\) have an \(\operatorname{Exp}\left(\frac{1}{2}\right)\) distribution. Determine the distribution function of \(\frac{1}{2} X\). What kind of distribution does \(\frac{1}{2} X\) have? b. Let \(X\) have an \(\operatorname{Exp}(\lambda)\) distribution. Determine the distribution function of \(\lambda X\). What kind of distribution does \(\lambda X\) have?

Let \(X\) be a continuous random variable with probability density function $$ f_{X}(x)= \begin{cases}\frac{3}{4} x(2-x) & \text { for } 0 \leq x \leq 2 \\\ 0 & \text { elsewhere }\end{cases} $$ a. Determine the distribution function \(F_{X}\). b. Let \(Y=\sqrt{X}\). Determine the distribution function \(F_{Y}\). c. Determine the probability density of \(Y\).

Let \(X\) be a continuous random variable with probability density \(f_{X}\) that takes only positive values and let \(Y=1 / X\). a. Determine \(F_{Y}(y)\) and show that $$ f_{Y}(y)=\frac{1}{y^{2}} f_{X}\left(\frac{1}{y}\right) \quad \text { for } y>0 . $$ b. Let \(Z=1 / Y\). Using a, determine the probability density \(f_{Z}\) of \(Z\), in terms of \(f_{X}\).

Let \(X\) have an \(\operatorname{Exp}(1)\) distribution, and let \(\alpha\) and \(\lambda\) be positive numbers. Determine the distribution function of the random variable $$ W=\frac{X^{1 / \alpha}}{\lambda} $$ The distribution of the random variable \(W\) is called the Weibull distribution with parameters \(\alpha\) and \(\lambda\).

Let \(X\) be a continuous random variable. Express the distribution function and probability density of the random variable \(Y=-X\) in terms of those of \(X\).

Let \(X\) be a random variable, and let \(g\) be a twice differentiable function with \(g^{\prime \prime}(x) \leq 0\) for all \(x\). Such a function is called a concave function. Show that for concave functions always $$ g(\mathrm{E}[X]) \geq \mathrm{E}[g(X)] $$

Let \(X\) be a random variable with the following probability mass function: \begin{tabular}{ccccc} \(x\) & 0 & 1 & 100 & 10000 \\ \hline \(\mathrm{P}(X=x)\) & \(\frac{1}{4}\) & \(\frac{1}{4}\) & \(\frac{1}{4}\) & \(\frac{1}{4}\) \end{tabular} a. Determine the distribution of \(Y=\sqrt{X}\). b. Which is larger \(\mathrm{E}[\sqrt{X}]\) or \(\sqrt{\mathrm{E}[X]}\) ? c. Compute \(\sqrt{\mathrm{E}[X]}\) and \(\mathrm{E}[\sqrt{X}]\) to check your answer (and to see that it makes a big difference!).

Let \(W\) have a \(U(\pi, 2 \pi)\) distribution. What is larger: \(\mathrm{E}[\sin (W)]\) or \(\sin (\mathrm{E}[W])\) ? Check your answer by computing these two numbers.

In this exercise we take a look at Jensen's inequality for the function \(g(x)=x^{3}(\) which is neither convex nor concave on \((-\infty, \infty)) .\) a. Can you find a (discrete) random variable \(X\) with \(\operatorname{Var}(X)>0\) such that $$ \mathrm{E}\left[X^{3}\right]=(\mathrm{E}[X])^{3} ? $$ b. Under what kind of conditions on a random variable \(X\) will the inequality \(\mathrm{E}\left[X^{3}\right]>(\mathrm{E}[X])^{3}\) certainly hold?

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be independent random variables, all with a \(U(0,1)\) distribution. Let \(Z=\max \left\\{X_{1}, \ldots, X_{n}\right\\}\) and \(V=\min \left\\{X_{1}, \ldots, X_{n}\right\\}\). a. Compute \(\mathrm{E}\left[\max \left\\{X_{1}, X_{2}\right\\}\right]\) and \(\mathrm{E}\left[\min \left\\{X_{1}, X_{2}\right\\}\right]\). b. Compute \(\mathrm{E}[Z]\) and \(\mathrm{E}[V]\) for general \(n\). c. Can you argue directly (using the symmetry of the uniform distribution (see Exercise 6.3) and not the result of the computation in b) that \(1-\mathrm{E}\left[\max \left\\{X_{1}, \ldots, X_{n}\right\\}\right]=\mathrm{E}\left[\min \left\\{X_{1}, \ldots, X_{n}\right\\}\right] ?\)

In this exercise we derive a kind of Jensen inequality for the minimum. a. Let \(a\) and \(b\) be real numbers. Show that $$ \min \\{a, b\\}=\frac{1}{2}(a+b-|a-b|) . $$ b. Let \(X\) and \(Y\) be independent random variables with the same distribution and finite expectation. Deduce from a that $$ \mathrm{E}[\min \\{X, Y\\}]=\mathrm{E}[X]-\frac{1}{2} \mathrm{E}[|X-Y|] $$ c. Show that $$ \mathrm{E}[\min \\{X, Y\\}] \leq \min \\{\mathrm{E}[X], \mathrm{E}[Y]\\} $$ Remark: this is not so interesting, since \(\min \\{\mathrm{E}[X], \mathrm{E}[Y]\\}=\mathrm{E}[X]=\mathrm{E}[Y]\), but we will see in the exercises of Chapter 11 that this inequality is also true for \(X\) and \(Y\), which do not have the same distribution.

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be independent random variables, all with an \(\operatorname{Exp}(\lambda)\) distribution. Let \(V=\min \left\\{X_{1}, \ldots, X_{n}\right\\}\). Determine the distribution function of \(V\). What kind of distribution is this?

From the "north pole" \(N\) of a circle with diameter 1 , a point \(Q\) on the circle is mapped to a point \(t\) on the line by its projection from \(N\), as illustrated in Figure \(8.2\). Suppose that the point \(Q\) is uniformly chosen on the circle. This is the same as saying that the angle \(\varphi\) is uniformly chosen from the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) (can you see this?). Let \(X\) be this angle, so that \(X\) is uniformly distributed over the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\). This means that \(\mathrm{P}(X \leq \varphi)=1 / 2+\varphi / \pi\) (cf. Quick exercise \(5.3)\). What will be the distribution of the projection of \(Q\) on the line? Let us call this random variable \(Z\). Then it is clear that the event \(\\{Z \leq t\\}\) is equal to the event \(\\{X \leq \varphi\\}\), where \(t\) and \(\varphi\) correspond to each other under the projection. This means that \(\tan (\varphi)=t\), which is the same as saying that \(\arctan (t)=\varphi .\) a. What part of the circle is mapped to the interval \([1, \infty)\) ? b. Compute the distribution function of \(Z\) using the correspondence between \(t\) and \(\varphi\). c. Compute the probability density function of \(Z\). The distribution of \(Z\) is called the Cauchy distribution (which will be discussed in Chapter 11).