We are given a continuous random variable $X$ with a probability density function (PDF) defined as:

$f(x) = \frac{1}{2} e^{-|x-1|}, \quad -\infty < x < \infty$

We are tasked with finding the probability $P(1 < |X| < 2)$.

Step 1: Break down the probability

We want to compute $P(1 < |X| < 2)$. This can be rewritten as:

$P(1 < |X| < 2) = P(-2 < X < -1 \, \text{or} \, 1 < X < 2)$

This is due to the absolute value condition. So, we now have two intervals of interest: $(-2, -1)$ and $(1, 2)$.

Thus, we can express the probability as:

$P(1 < |X| < 2) = P(-2 < X < -1) + P(1 < X < 2)$

Step 2: Find the cumulative probability for each region

Since we are working with a continuous probability density function, we compute these probabilities using integrals of the PDF over the respective intervals.

For $P(-2 < X < -1)$:

$P(-2 < X < -1) = \int_{-2}^{-1} \frac{1}{2} e^{-|x-1|} \, dx$

On this interval $(-2, -1)$, $|x-1| = -(x-1) = 1-x$, so the integral becomes:

$P(-2 < X < -1) = \int_{-2}^{-1} \frac{1}{2} e^{x-1} \, dx$

For $P(1 < X < 2)$:

$P(1 < X < 2) = \int_{1}^{2} \frac{1}{2} e^{-|x-1|} \, dx$

On this interval $(1, 2)$, $|x-1| = x-1$, so the integral becomes:

$P(1 < X < 2) = \int_{1}^{2} \frac{1}{2} e^{-(x-1)} \, dx$

Step 3: Solve the integrals

Now, let's compute both integrals.

Integral for $P(-2 < X < -1)$: