(a) Show by induction that if f: 𝐑 →𝐑 is convex then, for any x1, x2, …, xn and λ1, λ2, …, λn wi

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(a) Show by induction that if f: 𝐑 →𝐑 is convex then, for...

(a) Show by induction that if $f: \mathbf{R} \rightarrow \mathbf{R}$ is convex then, for any $x_{1}, x_{2}, \ldots, x_{n}$ and $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$ with $\sum_{i=1}^{n} \lambda_{i}=1$, $$ f\left(\sum_{i=1}^{n} \lambda_{i} x_{i}\right) \leq \sum_{i=1}^{n} \lambda_{i} f\left(x_{i}\right) $$ (b) Use Eqn. (2.2) to prove that if $f: \mathbf{R} \rightarrow \mathbf{R}$ is convex then $$ \mathbf{E}[f(X)] \geq f(\mathbf{E}[X]) $$ for any random variable $X$ that takes on only finitely many values.

(a) Show by induction that if $f: \mathbf{R} \rightarrow \mathbf{R}$ is convex then, for any $x_{1}, x_{2}, \ldots, x_{n}$ and $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$ with $\sum_{i=1}^{n} \lambda_{i}=1$,
$$
f\left(\sum_{i=1}^{n} \lambda_{i} x_{i}\right) \leq \sum_{i=1}^{n} \lambda_{i} f\left(x_{i}\right)
$$
(b) Use Eqn. (2.2) to prove that if $f: \mathbf{R} \rightarrow \mathbf{R}$ is convex then
$$
\mathbf{E}[f(X)] \geq f(\mathbf{E}[X])
$$
for any random variable $X$ that takes on only finitely many values.

Key Concepts

Convex Functions

A convex function is one where, for any two points in its domain, the function’s value at a weighted average of these points is less than or equal to the weighted average of the function’s values at these points. This property is fundamental in optimization, analysis, and probability, ensuring that the function lies below the chord connecting points on its graph.

Jensen's Inequality

Jensen's Inequality is a key result concerning convex functions which states that a convex function of an average is less than or equal to the average of the function’s values. It is widely used in probability theory and statistics to relate the expectation of a transformed random variable to the transformation of the variable’s expectation, providing bounds and insights into variability.

Mathematical Induction

Mathematical induction is a proof technique used to establish the validity of statements over an infinite sequence, usually the natural numbers. In the context of convex functions and Jensen’s inequality, induction is employed to extend the inequality from a base case (often two variables) to any finite number of variables, ensuring that the inequality holds generally.

Expectation of a Random Variable

The expectation of a random variable is a fundamental concept in probability representing the average or mean value the variable takes. It is central to many probabilistic inequalities, such as Jensen's inequality, where it is used to compare the average of a function of the random variable with the function evaluated at the average of the random variable.

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