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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 \( P(-2 < X < -1) \):

\[
\int_{-2}^{-1} \frac{1}{2} e^{x-1} \, dx = \frac{1}{2} \left[ e^{x-1} \right]_{-2}^{-1}
= \frac{1}{2} \left( e^{-2} - e^{-3} \right)
\]

#### Integral for \( P(1 < X < 2) \):

\[
\int_{1}^{2} \frac{1}{2} e^{-(x-1)} \, dx = \frac{1}{2} \left[ -e^{-(x-1)} \right]_{1}^{2}
= \frac{1}{2} \left( 1 - e^{-1} \right)
\]

### Step 4: Final computation

Now we combine the results:

\[
P(1 < |X| < 2) = \frac{1}{2} \left( e^{-2} - e^{-3} \right) + \frac{1}{2} \left( 1 - e^{-1} \right)
\]

This gives us the final value for the probability.

Would you like me to calculate the final numerical value for this probability or expand on any steps further?

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Here are 5 questions related to this problem:

1. What is the effect of absolute value conditions on probability intervals?
2. How do we adjust the probability density function when \( x \) is in a certain range?
3. What is the significance of the PDF normalization condition?
4. How do we interpret the probability of continuous random variables geometrically?
5. Why is it important to compute probabilities through integration for continuous distributions?

**Tip:** When handling absolute values in probability distributions, split the intervals based on where the expression inside the absolute value changes, then compute for each interval separately.

Let X be a continuous random variable with probability density function f(x) = 1/2 * e^{-|x-1|}, −∞ < x < ∞. Find the value of P(1 < |X| < 2).

Learn how to find the probability P(1 < |X| < 2) for a continuous random variable X with a given probability density function. This solution involves breaking down the probability into intervals, setting up integrals, and solving them step by step.

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