Because the expected value concept plays an important role in many economic
theories, it may be useful to summarize a few more properties of this
statistical measure. Throughout this problem, \(x\) is assumed to be a
continuous random variable with PDF \(f(x)\)
a. (Jensen's inequality) Suppose that \(g(x)\) is a concave function. Show that
\(E[g(x)] \leq g[E(x)] .\) Hint: Construct the tangent to \(g(x)\) at the point
\(E(x)\). This tangent will have the form \(c+d x \geq g(x)\) for all values of
\(x\) and \(c+d E(x)=g[E(x)],\) where \(c\) and \(d\) are constants.
b. Use the procedure from part (a) to show that if \(g(x)\) is a convex
function, then \(E[g(x)] \geq g[E(x)]\)
c. Suppose \(x\) takes on only non-negative values-that is, \(0 \leq x \leq
\infty\). Use integration by parts to show that
\\[
E(x)=\int[1-F(x)] d x
\\]
where \(F(x)\) is the cumulative distribution function for \(x\)
\\[
\left[\text { ice, } F(x)=\int_{0}^{x} f(t) d t\right]
\\]
d. (Markov's inequality) Show that if \(x\) takes on only positive values, then
the following inequality holds:
\\[
\begin{array}{c}
P(x \geq t) \leq \frac{E(x)}{t} \\
\text { Hint: } E(x)=\int_{0}^{\infty} x f(x) d x=\int_{0}^{t} x f(x) d
x+\int_{t}^{\infty} x f(x) d x
\end{array}
\\]
e. Consider the PDF \(f(x)=2 x^{-3}\) for \(x \geq 1\)
1\. Show that this is a proper PDF.
2\. Calculate \(F(x)\) for this PDF.
3\. Use the results of part (c) to calculate \(E(x)\) for this \(P D F\)
4\. Show that Markov's incquality holds for this function.
f. The concept of conditional expected value is useful in some economic
problems. We denote the expected value of \(x\) conditional on the occurrence of
some event, \(A\), as \(E(x | A),\) To compute this value we need to know the PDF
for \(x \text { given that } A \text { has occurred [denoted by } f(x | A)]\)
With this notation, \(E(x | A)=\int_{-\infty}^{+\infty} x f(x | A) d x .\)
Perhaps the easiest way to understand these relationships is with an example.
Let
\\[
f(x)=\frac{x^{2}}{3} \quad \text { for } \quad-1 \leq x \leq 2
\\]
1\. Show that this is a proper PDF.
2\. Calculate \(E(x)\)
3\. Calculate the probability that \(-1 \leq x \leq 0\)
4\. Consider the event \(0 \leq x \leq 2,\) and call this event \(A\). What is
\(f(x | A) ?\)
5\. Calculate \(E(x | A)\)
6\. Explain your results intuitively.
