# Segment 4. The Jailer's Tip

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## Contents

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### Problems

#### To Calculate

1. Evaluate $\int_0^1 \delta(3x-2) dx$

2. Prove that $\delta(a x) = \frac{1}{a}\delta(x)$.

3. What is the numerical value of $P(A|S_BI)$ if the prior for $p(x)$ is a massed prior with half the mass at $x = 1/3$ and half the mass at $x = 2/3$?

1. With respect to problem 3, above, since x is a probability, how can choosing x=1/3 half the time, and x=2/3 the other half of the time be different from choosing x=1/2 all the time?

2. Suppose A is some event that we view as stochastic with P(A), such as "will it rain today?". But the laws of physics (or meteorology) say that A actually depends on other weather variables X, Y, Z, etc., with conditional probabilities P(A|XYZ...). If we repeatedly sample just A, to naively measure P(A), are we correctly marginalizing over the other variables?