Jensen Gap for the Exponential

Problem

A random variable equals x1 with probability p and x2 otherwise. Return the Jensen gap for the convex function f(x) = e^x:

gap = E[e^X] - e^(E[X]) >= 0.

The gap is 0 when the two outcomes coincide.

Hint

Compute the expectation of f first, then subtract f of the expectation.

case 1

in: [0,1,0.5]

expected: 0.21041964352939435

case 2

in: [0,0,0.5]

expected: 0

case 3

in: [1,2,0.25]

expected: 0.4667598553070178

case 4

in: [-1,1,0.5]

expected: 0.5430806348152437

Starter codePython

import math


def jensen_gap_exponential(x1, x2, p):
    # Your code here
    pass

4 test cases · run your code against them in the editor