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Chapter 4: Q125SE (page 197)

A function g(x) is convex if the chord connecting any two points on the function鈥檚 graph lies above the graph. When g(x) is differentiable, an equivalent condition is that for every x, the tangent line at x lies entirely on or below the graph. (See the figure below.) How does \({\rm{E(g(X))}} \ge {\rm{g(E(X))}}\)compare to E(g(X))? (Hint: The equation of the tangent line at \({\rm{x = \mu }}\) is \({\rm{y = g(\mu ) + g'(\mu ) \times (x - \mu )}}\). Use the condition of convexity, substitute X for x, and take expected values. (Note: Unless g(x) is linear, the resulting inequality (usually called Jensen鈥檚 inequality) is strict (\({\rm{ < }}\) rather than \( \le \)); it is valid for both continuous and discrete rv鈥檚.)

Short Answer

Expert verified

(a) \({\rm{E(g(x)) = g(\mu )}}\) and \({\rm{V(g(x)) = }}{\left( {{\rm{g'(\mu )}}} \right)^{\rm{2}}}{\rm{ \times }}{{\rm{\sigma }}^{\rm{2}}}\) is expressions for these approximately.

(b) \({{\rm{\mu }}_{\rm{R}}}{\rm{ = }}\frac{{\rm{v}}}{{{\rm{20}}}}\) and \({{\rm{\sigma }}_{\rm{R}}}{\rm{ = }}\frac{{\rm{v}}}{{{\rm{800}}}}\).

Step by step solution

01

Definition of Tangent line

At a given point, the tangent line (or simply tangent) to a plane curve is the straight line that "just touches" the curve.

02

Determining comparison of \({\rm{E(g(X))}} \ge {\rm{g(E(X))}}\) to E(g(x))

The equation of the line tangent to the graph of \({\rm{g(x)}}\) at \({\rm{x = \mu }}\) is as follows:

\({\rm{y = g(\mu ) + g'(\mu ) \times (x - \mu )}}\)

The convexity requirement states that a function is convex if the tangent line at x lies wholly on or below the graph of g(x) for every x. As a result, we can write:

\({\rm{g(x)}} \ge {\rm{g(\mu ) + g'(\mu ) \times (x - \mu )}}\)

Then we replace x with X.

\({\rm{E(g(X))}} \ge {\rm{E}}\left( {{\rm{g(\mu ) + g'(\mu ) \times (X - \mu )}}} \right)\)

\({\rm{E(g(X))}} \ge {\rm{E(g(\mu )) + E}}\left( {{\rm{g'(\mu ) \times (X - \mu )}}} \right)\)

\({\rm{E(g(X))}} \ge {\rm{g(\mu ) + g'(\mu ) \times (E(X - \mu ))}}\)

\({\rm{E(g(X))}} \ge {\rm{g(\mu ) + g'(\mu ) \times (E(X) - E(\mu ))}}\)

\({\rm{E(g(X))}} \ge {\rm{g(\mu ) + g'(\mu ) \times (\mu - \mu )}}\)

\({\rm{E(g(X))}} \ge {\rm{g(\mu )}}\)

As a result, if g(x) is convex, then: \({\rm{E(g(X))}} \ge {\rm{g(E(X))}}\)

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