A continuous RV X has density function 
f(x)={(1/(1+x)^2,x≥0@0,otherwise)
Calculate P(2<X<5)

We are given the probability density function (PDF) of a continuous random variable \( X \), defined as:

\[
f(x) = 
\begin{cases}
\frac{1}{(1+x)^2}, & x \geq 0 \\
0, & \text{otherwise}
\end{cases}
\]

We are asked to find \( P(2 < X < 5) \), which is the probability that \( X \) lies between 2 and 5. This probability can be computed by integrating the PDF over the interval \( (2, 5) \):

\[
P(2 < X < 5) = \int_{2}^{5} f(x) \, dx
\]

Substituting the PDF:

\[
P(2 < X < 5) = \int_{2}^{5} \frac{1}{(1+x)^2} \, dx
\]

To solve this, we perform the following integration.

### Step-by-step solution:

1. Let’s use a substitution to simplify the integral:
   - Let \( u = 1 + x \), so \( du = dx \).
   - When \( x = 2 \), \( u = 3 \), and when \( x = 5 \), \( u = 6 \).

Now, the integral becomes:

\[
P(2 < X < 5) = \int_{3}^{6} \frac{1}{u^2} \, du
\]

2. The integral of \( \frac{1}{u^2} \) is:

\[
\int \frac{1}{u^2} \, du = -\frac{1}{u}
\]

Thus, the integral evaluates to:

\[
P(2 < X < 5) = \left[ -\frac{1}{u} \right]_{3}^{6}
\]

3. Substituting the limits:

\[
P(2 < X < 5) = -\frac{1}{6} + \frac{1}{3}
\]

4. Simplifying:

\[
P(2 < X < 5) = \frac{1}{3} - \frac{1}{6} = \frac{1}{6}
\]

Thus, the probability that \( X \) lies between 2 and 5 is:

\[
P(2 < X < 5) = \frac{1}{6}
\]

---

Would you like further details or have any questions?

Here are 5 related questions:
1. How would we compute \( P(X > 5) \) for this random variable?
2. Can you generalize the process for finding \( P(a < X < b) \) for any \( a \) and \( b \)?
3. What is the cumulative distribution function (CDF) of this random variable?
4. How would you compute the mean of this random variable?
5. Can you find the variance of this random variable?

**Tip**: When solving integrals involving rational functions, substitution is often a very useful technique for simplification!

A continuous RV X has density function f(x)={(1/(1+x)^2, x≥0; 0, otherwise). Calculate P(2<X<5).

Learn how to calculate the probability P(2 < X < 5) for a continuous random variable X with the probability density function f(x) = 1/(1+x)^2 for x ≥ 0. This step-by-step guide covers integration techniques and substitution methods. Suitable for college-level probability and statistics students.

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