# Travis: Segment 4

#### To Calculate

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

$u = 3x-2 \to du = 3dx$

$\int_0^1 \delta(3x-2) dx \to \frac{1}{3} \int_{-2}^1 \delta(u) du = \frac{1}{3}$

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

$u = ax \to du = adx$

$\int \delta(ax) dx \to \frac{1}{a} \int \delta(u) du = \frac{1}{a}$

$\therefore \delta(a x) = \frac{1}{a}\delta(x)$

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 $x = 2/3$?

$P(A|S_BI) = \frac{1}{2} \int_0^1\frac{1}{1+x}\delta\Bigg(x-\frac{1}{3}\Bigg)dx + \frac{1}{2}\int_0^1\frac{1}{1+x}\delta\Bigg(x-\frac{2}{3}\Bigg)dx$

$P(A|S_BI) = \frac{1}{2}\Bigg(\frac{1}{1+\frac{1}{3}}\Bigg) + \frac{1}{2}\Bigg(\frac{1}{1+\frac{2}{3}}\Bigg) = \frac{1}{2}\Bigg(\frac{3}{4}\Bigg) + \frac{1}{2}\Bigg(\frac{3}{5}\Bigg) = \frac{3}{8} + \frac{3}{10} = \frac{27}{40}$