Refer to the random variable X defined in Exercise 5.6.2. Compute the - f(x) =
⎧ ⎪ ⎨ ⎪ ⎩
0 if x < 1
2(x− 1) if 1 ≤ x ≤ 2
0 if x > 2
⎫ ⎪ ⎬ ⎪ ⎭.
following two quantities: q2(X), the population median; and iqr(X),
the population interquartile range.

We are asked to compute the population median \( q_2(X) \) and the interquartile range (IQR) for the random variable \( X \), whose probability density function \( f(x) \) is defined as follows:

\[
f(x) = \begin{cases}
0 & \text{if } x < 1, \\
2(x - 1) & \text{if } 1 \leq x \leq 2, \\
0 & \text{if } x > 2.
\end{cases}
\]

### Step 1: Verify that \( f(x) \) is a valid probability density function

To ensure that \( f(x) \) is a valid probability density function, we need to check if the total area under the curve equals 1:

\[
\int_{-\infty}^{\infty} f(x) \, dx = 1.
\]

Since \( f(x) = 0 \) for \( x < 1 \) and \( x > 2 \), we only need to integrate from 1 to 2:

\[
\int_1^2 2(x - 1) \, dx.
\]

We will compute this integral.

\[
\int 2(x - 1) \, dx = (x - 1)^2.
\]

Now, evaluate the definite integral from 1 to 2:

\[
\left[ (x - 1)^2 \right]_1^2 = (2 - 1)^2 - (1 - 1)^2 = 1.
\]

Thus, the total area is 1, confirming that \( f(x) \) is a valid probability density function.

### Step 2: Compute the median \( q_2(X) \)

The median \( q_2(X) \) is the value of \( m \) such that:

\[
P(X \leq m) = 0.5.
\]

This means we need to find \( m \) such that:

\[
\int_1^m 2(x - 1) \, dx = 0.5.
\]

We already know the antiderivative is \( (x - 1)^2 \). So we set up the equation:

\[
\left[ (x - 1)^2 \right]_1^m = 0.5.
\]

Evaluating the integral, we get:

\[
(m - 1)^2 = 0.5.
\]

Taking the square root of both sides:

\[
m - 1 = \sqrt{0.5}.
\]

Thus,

\[
m = 1 + \sqrt{0.5} = 1 + \frac{\sqrt{2}}{2} \approx 1.707.
\]

So, the population median is approximately:

\[
q_2(X) \approx 1.707.
\]

### Step 3: Compute the interquartile range (IQR)

The interquartile range (IQR) is the difference between the third quartile \( q_3(X) \) and the first quartile \( q_1(X) \). The first quartile \( q_1(X) \) is defined by:

\[
P(X \leq q_1) = 0.25,
\]

and the third quartile \( q_3(X) \) is defined by:

\[
P(X \leq q_3) = 0.75.
\]

#### Finding \( q_1(X) \):

We solve for \( q_1 \) such that:

\[
\int_1^{q_1} 2(x - 1) \, dx = 0.25.
\]

As before, we use the antiderivative \( (x - 1)^2 \):

\[
(q_1 - 1)^2 = 0.25.
\]

Taking the square root of both sides:

\[
q_1 - 1 = \sqrt{0.25} = 0.5,
\]

so

\[
q_1 = 1.5.
\]

#### Finding \( q_3(X) \):

We solve for \( q_3 \) such that:

\[
\int_1^{q_3} 2(x - 1) \, dx = 0.75.
\]

We again use the antiderivative:

\[
(q_3 - 1)^2 = 0.75.
\]

Taking the square root of both sides:

\[
q_3 - 1 = \sqrt{0.75} = \frac{\sqrt{3}}{2} \approx 0.866.
\]

Thus,

\[
q_3 = 1 + \frac{\sqrt{3}}{2} \approx 1.866.
\]

#### Computing the IQR:

The interquartile range is:

\[
\text{IQR} = q_3 - q_1 = 1.866 - 1.5 = 0.366.
\]

### Final Results:
- The population median \( q_2(X) \approx 1.707 \).
- The interquartile range (IQR) \( \approx 0.366 \).

Would you like more details on any of these steps, or help with a related topic?

### 5 Related Questions:
1. How do you compute the cumulative distribution function (CDF) from a probability density function (PDF)?
2. What are the properties of a valid probability density function?
3. How can you interpret the median and interquartile range in terms of probability?
4. What is the relationship between the IQR and the spread of data in a distribution?
5. How can we extend these concepts to discrete random variables?

### Tip:
The IQR is a robust measure of variability since it is less affected by extreme values compared to the standard deviation.

Refer to the random variable X defined in Exercise 5.6.2. Compute the following two quantities: q2(X), the population median; and iqr(X), the population interquartile range.

Learn how to calculate the population median (q2(X)) and interquartile range (IQR) for the random variable X with a given probability density function. This solution involves applying integration and understanding key probability concepts. Suitable for undergraduate students in mathematics or statistics.

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