# Segment 4: The Jailer's Tip - 1/25/2013

**Problem 1**

First, we need to make our integral into the form of <math>\int_{-\infty}^\infty \delta(u)du</math>

<math>\int_0^1 \delta(3x-2) dx</math>

Let u = 3x - 2

du = 3dx

dx = du/3

To change our bounds into terms of u:

- For 0: 3(0) - 2 = -2
- For 1: 3(1) - 2 = 1

We know that <math>\int_{-\infty}^\infty \delta(u)du = 1</math> since only at 0 we get 1. Thus, our integral becomes

<math>\frac{1}{3}\cdot \int_{-2}^1 \delta(u) du = 1 \cdot \frac{1}{3} = \frac{1}{3} </math>

**Problem 2:** Prove that <math> \delta(ax) = \frac{1}{a}\delta(x) </math>

<math>\int_{-\infty}^\infty \delta(ax)du</math>

Let u = ax

du = a dx

dx = du/a

<math>\frac{1}{a}\int_{-\infty}^\infty \delta(u)du</math> = <math>\bold{\frac{1}{a}}</math>

<math>\int_{-\infty}^\infty \frac{1}{a}\delta(x) = \frac{1}{a}\int_{-\infty}^\infty \delta(x) </math> = <math>\bold{\frac{1}{a}}</math>

**Problem 3: **

- <math>\int_0^1 \frac{1}{1 + x } \cdot \frac{1}{2} \cdot (\delta(x - \frac{1}{3} )+ \delta(x - \frac{2}{3})dx)</math>
- <math>\frac{1}{2}\cdot (\frac{1}{1 + \frac{1}{3}} + \frac{1}{1 + \frac{2}{3}})</math>
- <math>\frac{1}{2}\cdot (\frac{1}{\frac{4}{3}} + \frac{1}{\frac{5}{3}})</math>
- <math>\frac{1}{2}\cdot (\frac{3}{4} + \frac{3}{5})</math>
- <math>\frac{1}{2}\cdot (\frac{15}{20} + \frac{12}{20})</math>
- <math>\frac{1}{2}\cdot (\frac{27}{20})</math>
- <math>\frac{27}{40} = 0.675 </math>

**To Think About 1: **

It should be the idea that we are no longer choosing a person uniformly. Even if we are choosing x=1/3 half the time and x=2/3 the other half, the overall probability is represent by the outcome of the probability, not a matter of us picking the probability uniformly. Since we have increased the probability of choosing one jailer over the other, it increases the likelihood of someone being chosen more often.

**To Think About 2**

Marginalizing is very similar to averaging. By sampling over A, A is being influenced by the different variables that affect the probability of A. After taking many samples of A, the Factors of X, Y,Z, etc. will be incorporated into the probability.