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

From Computational Statistics (CSE383M and CS395T)
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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.