Problem 189 - An Introduction to Mathematical Statistics and Its Applications

Question

Let \(Y\) be a continuous nonnegative random variable. Show that \(W=Y^{2}\) has pdf \(f_{W}(w)=\frac{1}{2 \sqrt{w}} f_{Y}(\sqrt{w})\). (Hint: First find \(F_{W}(w) .\) )

Step-by-Step Solution

Answer

The pdf of the random variable \(W = Y^{2}\) is derived to be \[f_{W}(w) = \frac{f_{Y}(\sqrt{w})}{2\sqrt{w}}\], where \(f_{W}(w)\) and \(f_{Y}(w)\) are the pdfs of \(W\) and \(Y\) respectively.

Step 1: Deriving the cumulative distribution function (CDF) of W

To find the pdf of \(W\), the CDF of \(W\) is first found. The CDF of \(W\), \(F_{W}(w)\), where \(W\) is defined as \(W = Y^{2}\), is obtained by solving: \[F_{W}(w) = P(W \leq w) = P(Y^{2} \leq w)\] Given that \(Y\) is a nonnegative random variable, square root can be applied without loss of generality on both sides, it becomes, \[P(Y \leq \sqrt{w})\] This is actually the CDF of \(Y\) at \(\sqrt{w}\), i.e., \(F_{W}(w) = F_{Y}(\sqrt{w})\].

Step 2: Deriving the pdf of W

By differentiating the CDF, the pdf is obtained. So, the pdf of \(W\), \(f_{W}(w)\) is: \[f_{W}(w) = \frac{d}{dw}F_{W}(w) = \frac{d}{dw}F_{Y}(\sqrt{w})\] By employing the chain rule for differentiation, \[f_{W}(w) = \frac{d}{dw}F_{Y}(\sqrt{w}) = f_{Y}(\sqrt{w}) \cdot \frac{1}{2\sqrt{w}}\]

Step 3: Verifying obtained pdf of W

Now, verifying the derived pdf, \[f_{W}(w) = \frac{f_{Y}(\sqrt{w})}{2\sqrt{w}}\] The obtained pdf is equivalent to what was required to show in the original problem which confirms the solution.

Key Concepts

Cumulative Distribution Function

The Cumulative Distribution Function (CDF) of a random variable, like our variable \(W\), plays a crucial role in understanding the probability distribution. The CDF provides the probability that a random variable is less than or equal to a certain value. In our exercise, we want to find \(F_{W}(w)\), the CDF of \(W\). This involves finding the probability that \(W\) is less than or equal to \(w\), expressed as \(P(W \leq w)\).

Since \(W = Y^2\), this probability translates into \(P(Y^2 \leq w)\). To make progress, we utilize the fact that \(Y\) is nonnegative, allowing us to take the square root of both sides, resulting in \(P(Y \leq \sqrt{w})\). This equation might look familiar—it’s essentially the CDF of \(Y\) evaluated at \(\sqrt{w}\), or \(F_{Y}(\sqrt{w})\).

Thus, the CDF of \(W\) is expressed as:

• \(F_{W}(w) = F_{Y}(\sqrt{w})\)

Understanding this relationship helps us move towards calculating the Probability Density Function (PDF) of the transformed variable \(W\).

Random Variables

Random variables are foundational in probability and statistics. A random variable is essentially a variable whose possible values are outcomes of a random phenomenon. There are different types of random variables; in our exercise, we deal with continuous and nonnegative ones.

Let’s consider \(Y\) in our exercise. Since \(Y\) is a continuous random variable, it can take any value in a given range with probabilities described by its Probability Density Function (PDF), \(f_{Y}(y)\). The nonnegativity of \(Y\) simplifies operations that involve roots and powers, as these functions are well-defined and single-valued for nonnegative numbers.

The transformation \(W = Y^2\) represents a new random variable based on \(Y\). This step is about understanding how transformations affect the distribution properties and outcome probabilities of the new variable \(W\). To manage this, we delve into the CDF and PDF of \(W\), employing techniques like differentiation and chain rule. This helps describe how the likelihoods distribute over the possible values \(W\) can take.

Chain Rule in Calculus

The chain rule is an essential concept in calculus, particularly when dealing with composite functions. It provides a systematic way to differentiate such functions. In our exercise, the difference in variable transformation highlights the utility of the chain rule.

Consider the transition from the Cumulative Distribution Function \(F_{W}(w)\) to the Probability Density Function (PDF) \(f_{W}(w)\). To make this leap, we differentiate \(F_{Y}(\sqrt{w})\) with respect to \(w\). This seemingly complex derivative becomes manageable with the chain rule, which accounts for the inner function \(\sqrt{w}\) within the larger function.

Here’s how it works:

• Differentiation of \(F_{Y}(\sqrt{w})\) with respect to \(w\) entails first taking the derivative of \(F_{Y}\) with respect to its argument (\(\sqrt{w}\))
• Multiply by the derivative of the inner function \(\sqrt{w}\) itself with respect to \(w\), which is \(\frac{1}{2\sqrt{w}}\)

Thus, using the chain rule, we determine:

• \(f_{W}(w) = f_{Y}(\sqrt{w}) \cdot \frac{1}{2\sqrt{w}}\)

This relationship effectively connects the PDF of \(Y\) to \(W\), showing how the transformation modifies probability density across different variables.