# proof of Jensen’s inequality

We prove an equivalent^{}, more convenient formulation: Let $X$ be some random variable^{}, and let $f(x)$ be a convex function (defined at least on a segment containing the range of $X$). Then the expected value of $f(X)$ is at least the value of $f$ at the mean of $X$:

$$\mathbb{E}[f(X)]\ge f(\mathbb{E}[X]).$$ |

Indeed, let $c=\mathbb{E}[X]$. Since $f(x)$ is convex, there exists a supporting line for $f(x)$ at $c$:

$$\phi (x)=\alpha (x-c)+f(c)$$ |

for some $\alpha $, and $\phi (x)\le f(x)$. Then

$$\mathbb{E}[f(X)]\ge \mathbb{E}[\phi (X)]=\mathbb{E}[\alpha (X-c)+f(c)]=f(c)$$ |

as claimed.

Title | proof of Jensen’s inequality |
---|---|

Canonical name | ProofOfJensensInequality |

Date of creation | 2013-03-22 12:45:15 |

Last modified on | 2013-03-22 12:45:15 |

Owner | Andrea Ambrosio (7332) |

Last modified by | Andrea Ambrosio (7332) |

Numerical id | 6 |

Author | Andrea Ambrosio (7332) |

Entry type | Proof |

Classification | msc 26D15 |

Classification | msc 39B62 |