Understanding the Interquartile Range (IQR) for a Random Variable

The interquartile range (IQR) is a measure of statistical dispersion, representing the spread of the middle 50% of the data. For a continuous random variable, it is the difference between the third quartile ($Q_3$) and the first quartile ($Q_1$).

• The first quartile ($Q_1$) is the value below which 25% of the probability mass lies. If $X$ is the random variable, $P(X \le Q_1) = 0.25$.
• The third quartile ($Q_3$) is the value below which 75% of the probability mass lies. $P(X \le Q_3) = 0.75$.
• The interquartile range (IQR) is calculated as $IQR = Q_3 - Q_1$.

To find the quartiles for a continuous random variable with a given probability density function (PDF), we first need to find its cumulative distribution function (CDF).

Calculating the Cumulative Distribution Function (CDF)

The given probability density function (PDF) is $f(x) = 4x^3$ for $0 < x < 1$. The CDF, $F(x)$, is the integral of the PDF from the lower limit of the support to $x$.

For $0 < x < 1$, the CDF is:

\( F(x) = P(X \le x) = \int_{0}^{x} f(t) dt \)

\( F(x) = \int_{0}^{x} 4t^3 dt \)

Integrating $4t^3$ with respect to $t$, we get \(t^4\). Evaluating this from 0 to $x$:

\( F(x) = \left[ t^4 \right]_{0}^{x} = x^4 - 0^4 = x^4 \)

So, the CDF is \(F(x) = x^4\) for $0 < x < 1$. The complete CDF is:

\( F(x) = \begin{cases} 0 & \text{for } x \le 0 \\ x^4 & \text{for } 0 < x < 1 \\ 1 & \text{for } x \ge 1 \end{cases} \)

Finding the Quartiles ($Q_1$ and $Q_3$)

To find the quartiles, we set the CDF equal to the desired probability and solve for $x$.

Finding the First Quartile ($Q_1$)

We need to find $Q_1$ such that \(F(Q_1) = 0.25\).

Using the CDF for $0 < x < 1$:

\( Q_1^4 = 0.25 \)

\( Q_1^4 = \dfrac{1}{4} \)

Taking the fourth root of both sides (and since $Q_1$ must be positive as per the domain):

\( Q_1 = \left( \dfrac{1}{4} \right)^{\dfrac{1}{4}} \)

Finding the Third Quartile ($Q_3$)

We need to find $Q_3$ such that \(F(Q_3) = 0.75\).

Using the CDF for $0 < x < 1$:

\( Q_3^4 = 0.75 \)

\( Q_3^4 = \dfrac{3}{4} \)

Taking the fourth root of both sides (and since $Q_3$ must be positive as per the domain):

\( Q_3 = \left( \dfrac{3}{4} \right)^{\dfrac{1}{4}} \)

Calculating the Interquartile Range (IQR)

The interquartile range (IQR) is \(Q_3 - Q_1\).

\( IQR = \left( \dfrac{3}{4} \right)^{\dfrac{1}{4}} - \left( \dfrac{1}{4} \right)^{\dfrac{1}{4}} \)

This matches one of the given options.

Summary of Interquartile Range Calculation

Here's a quick summary of the steps:

1. Find the CDF, \(F(x)\), from the given PDF, \(f(x)\).
2. Set \(F(x) = 0.25\) and solve for \(x\) to find \(Q_1\).
3. Set \(F(x) = 0.75\) and solve for \(x\) to find \(Q_3\).
4. Calculate \(IQR = Q_3 - Q_1\).

Revision Table: Key Concepts for IQR

Additional Information on Percentiles and IQR

Quartiles are specific types of percentiles. The first quartile ($Q_1$) is the 25th percentile, the second quartile ($Q_2$, which is also the median) is the 50th percentile, and the third quartile ($Q_3$) is the 75th percentile. The IQR is a robust measure of spread because it is not affected by extreme outliers, unlike the range or standard deviation.

For any continuous random variable with PDF \(f(x)\), if the support is \([a, b]\), the CDF is \(F(x) = \int_{a}^{x} f(t) dt\). To find the percentile \(P_p\) (where \(p\) is a probability between 0 and 1), you solve \(F(P_p) = p\). The quartiles are special cases where \(p\) is 0.25, 0.50, and 0.75.