Problem 93 - An Introduction to Mathematical Statistics and Its Applications

Question

Show that a gamma pdf has the unique mode \(\frac{r-1}{\lambda}\); that is, show that the function \(f_{Y}(y)=\frac{\alpha^{\prime}}{\Gamma(r)} y^{\prime-1} e^{-\lambda y}\) takes its maximum value at \(y_{\text {mode }}=\frac{r-1}{\lambda}\) and at no other point.

Step-by-Step Solution

Answer

To show the Gamma pdf has a unique mode at \(y_{\text {mode}}=\frac{r-1}{\lambda}\), differentiate the function and set it equal to zero. Solve for y to get the critical points. Verify that the unique mode is \(\frac{r-1}{\lambda}\) by substituting it into the locations where y equals the solution from the differentiation.

Step 1: Identify the function and mode

The Gamma pdf is given as \(f_{Y}(y)=\frac{\alpha^{\prime}}{\Gamma(r)} y^{\prime-1} e^{-\lambda y}\). The mode which we expect the function to achieve its maximum is \(y_{\text {mode}}=\frac{r-1}{\lambda}\).

Step 2: Differentiate the Gamma pdf

Differentiate the function with respect to y. Noting that the derivative of \(y^{\prime-1}\) is \((\prime-1)y^{\prime-2}\) and the derivative of \(e^{-\lambda y}\) is \(-\lambda e^{-\lambda y}\), the derivative of the function \(f_{Y}(y)\) can be found.

Step 3: Find the critical points

Set the derivative of the function to zero and solve for y. This gives the critical points of the function.

Step 4: Verify the mode

Substitute \(\frac{r-1}{\lambda}\) into the places where y equals the solution from step 3. If it is true, then it shows that the function reaches its maximum at \(y_{\text {mode}}=\frac{r-1}{\lambda}\) and at no other point.

Key Concepts

Probability Density Function

In the realm of statistics, the **Probability Density Function** (PDF) is fundamental when dealing with continuous random variables. A PDF specifies the likelihood of a given random variable falling within a specific range. For the Gamma Distribution, the PDF is given by the formula:\[f_{Y}(y) = \frac{\alpha^{'}}{\Gamma(r)} y^{\prime-1} e^{-\lambda y}\]Here:

• \(\alpha^{'}\) and \(\Gamma(r)\) are constants.
• \(y^{\prime-1}\) represents raising \(y\) to the power minus one.
• \(e^{-\lambda y}\) is an exponential decay based on the parameter \(\lambda\).

It's important to remember that the area under a PDF over its entire range equals one, implying the certainty of some outcome within the scope of the variable's distribution. Unlike discrete probability mass functions, PDFs do not give probabilities directly at specific points but over intervals.

Mode of a Distribution

The **Mode of a Distribution** is a well-known statistical measure that indicates the value at which a probability distribution attains its highest peak. In simpler terms, for a continuous distribution like the Gamma distribution, it's where the PDF is at its greatest.When exploring the Gamma distribution, we are interested in finding the mode of the PDF, which effectively captures the most "likely" outcome. In our case, this mode is theoretically predicted to be:\[y_{\text{mode}} = \frac{r-1}{\lambda}\]This formula suggests that as the shape parameter \(r\) increases, the mode also shifts to the right, provided that \(\lambda\) remains constant. Understanding the mode helps in summarizing the central tendency of the distribution.

Differentiation in Calculus

**Differentiation in Calculus** is a powerful tool used to explore functions, including determining maxima and minima. It involves computing the derivative of a function with respect to an independent variable to see how the function behaves as that variable changes.In the exercise related to the Gamma distribution, differentiation is used to find the derivative of the Gamma PDF. By differentiating:\[f_{Y}(y) = \frac{\alpha^{'}}{\Gamma(r)} y^{\prime-1} e^{-\lambda y}\]We look for where this derivative equals zero, as these points suggest potential maxima or minima, known as critical points. The first derivative tells us about the function's slope, and a zero-value derivative often indicates a turning point, useful in identifying modes, as we are doing here.

Critical Points in Functions

**Critical Points in Functions** are vital in understanding the behavior of functions. They occur where the derivative of a function is zero or undefined, and they may indicate places where the function reaches a local maximum, a local minimum, or a saddle point.In the context of the Gamma PDF, finding critical points involves setting the derivative to zero and solving for \(y\). These critical points are candidates for the mode of the distribution, where the function could potentially achieve its maximum value.In our problem, the critical point\[y_{\text{critical}} = \frac{r-1}{\lambda}\]matches our predicted mode. By verifying this point against our PDF, we confirm it achieves the maximum value at this particular point and thus identify it as the distribution's mode. This verification emphasizes the importance of critical points in assessing distribution properties.