Shortcut For Double-integral Over A Triangle

Introduction

In probability theory and statistics, double-integrals over a triangle are a crucial concept in evaluating the joint distribution of two random variables. When dealing with two independent but not identically distributed random variables, $X$ and $Y$, on the interval $[0,1]$, the joint distribution function $f_{X,Y}(x,y)$ can be expressed as a double-integral over a triangular region. In this article, we will explore a shortcut for evaluating this double-integral, making it easier to compute the joint distribution of $X$ and $Y$.

Joint Distribution Function

The joint distribution function of two random variables $X$ and $Y$ is given by:

$$f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y)$$

where $f_X(x)$ and $f_Y(y)$ are the marginal distribution functions of $X$ and $Y$, respectively. However, when the random variables are not identically distributed, the joint distribution function can be expressed as a double-integral over a triangular region:

$$f_{X,Y}(x,y) = \int_{0}^{x} \int_{0}^{y} f_{X,Y}(u,v) \, du \, dv$$

Double-Integral over a Triangle

To evaluate the double-integral over a triangle, we need to integrate the joint distribution function $f_{X,Y}(x,y)$ over the triangular region bounded by the lines $x=0$, $y=0$, and $x+y=1$. This can be done using the following formula:

$$\int_{0}^{1} \int_{0}^{1-x} f_{X,Y}(x,y) \, dy \, dx$$

However, this formula can be computationally intensive, especially when dealing with complex joint distribution functions.

Shortcut for Double-Integral

A shortcut for evaluating the double-integral over a triangle is to use the following formula:

$$\int_{0}^{1} \int_{0}^{1-x} f_{X,Y}(x,y) \, dy \, dx = \int_{0}^{1} f_X(x) \cdot \left( \int_{0}^{1-x} f_Y(y) \, dy \right) \, dx$$

This formula allows us to separate the double-integral into two single-integrals, making it easier to compute the joint distribution of $X$ and $Y$.

Derivation of the Shortcut Formula

To derive the shortcut formula, we can start by writing the double-integral over a triangle as:

$$\int_{0}^{1} \int_{0}^{1-x} f_{X,Y}(x,y) \, dy \, dx = \int_{0}^{1} \int_{0}^{1-x} f_X(x) \cdot f_Y(y) \, dy \, dx$$

Using the substitution $u = 1-x$, we can rewrite the double-integral as:

$$\int_{0}^{1} \int_{0}^{1-x} f_X(x) \cdot f_Y(y) \, dy \, dx = \int_{0}^{1} f_X(x) \cdot \left( \int_{0}^{1-x} f_Y(y) \, dy \right) \, dx$$

This is the shortcut formula for evaluating the double-integral over a triangle.

Example

Suppose we have two random variables $X$ and $Y$ that are independently but not identically distributed on $[0,1]$ according to the following marginal distribution functions:

$$f_X(x) = x^2$$

$$f_Y(y) = 2y$$

We can use the shortcut formula to evaluate the joint distribution function of $X$ and $Y$:

$$f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y) = x^2 \cdot 2y = 2x^2y$$

Using the shortcut formula, we can evaluate the double-integral over a triangle as:

$$\int_{0}^{1} \int_{0}^{1-x} f_{X,Y}(x,y) \, dy \, dx = \int_{0}^{1} f_X(x) \cdot \left( \int_{0}^{1-x} f_Y(y) \, dy \right) \, dx$$

$$= \int_{0}^{1} x^2 \cdot \left( \int_{0}^{1-x} 2y \, dy \right) \, dx$$

= \int_{0}^{1} x^2 \cdot \left( y^2 \right|_{0}^{1-x} \right) \, dx

$$= \int_{0}^{1} x^2 \cdot (1-x)^2 \, dx$$

$$= \int_{0}^{1} x^2 - 2x^3 + x^4 \, dx$$

$$= \left. \frac{x^3}{3} - \frac{2x^4}{4} + \frac{x^5}{5} \right|_{0}^{1}$$

$$= \frac{1}{3} - \frac{1}{2} + \frac{1}{5}$$

$$= \frac{10-15+6}{30}$$

$$= \frac{1}{30}$$

Conclusion

Q: What is the shortcut formula for evaluating the double-integral over a triangle?

A: The shortcut formula is:

$$\int_{0}^{1} \int_{0}^{1-x} f_{X,Y}(x,y) \, dy \, dx = \int_{0}^{1} f_X(x) \cdot \left( \int_{0}^{1-x} f_Y(y) \, dy \right) \, dx$$

Q: How do I apply the shortcut formula to evaluate the joint distribution function of two random variables?

A: To apply the shortcut formula, you need to:

1. Identify the marginal distribution functions of the two random variables, $f_X(x)$ and $f_Y(y)$.
2. Substitute these functions into the shortcut formula.
3. Evaluate the resulting double-integral.

Q: What are the assumptions of the shortcut formula?

A: The shortcut formula assumes that the two random variables, $X$ and $Y$, are independently but not identically distributed on the interval $[0,1]$.

Q: Can I use the shortcut formula for other types of integrals?

A: No, the shortcut formula is specifically designed for evaluating double-integrals over a triangle. It may not be applicable to other types of integrals.

Q: How do I handle cases where the joint distribution function is not separable?

A: If the joint distribution function is not separable, you may need to use other methods, such as numerical integration or approximation techniques, to evaluate the double-integral.

Q: Can I use the shortcut formula for continuous random variables with different support intervals?

A: No, the shortcut formula is specifically designed for continuous random variables with support intervals $[0,1]$. You may need to modify the formula or use other methods for random variables with different support intervals.

Q: Are there any limitations to the shortcut formula?

A: Yes, the shortcut formula has several limitations, including:

• It assumes that the two random variables are independently but not identically distributed.
• It is specifically designed for double-integrals over a triangle.
• It may not be applicable to other types of integrals or random variables.

Q: Can I use the shortcut formula for discrete random variables?

A: No, the shortcut formula is specifically designed for continuous random variables. You may need to use other methods, such as numerical integration or approximation techniques, to evaluate the double-integral for discrete random variables.

Q: How do I verify the accuracy of the shortcut formula?

A: To verify the accuracy of the shortcut formula, you can:

1. Compare the result of the shortcut formula with the exact result of the double-integral.
2. Use numerical integration or approximation techniques to evaluate the double-integral and compare the result with the shortcut formula.
3. Check the assumptions of the shortcut formula to ensure that they are met.

Q: Can I use the shortcut formula for other applications in probability theory and statistics?

A: Yes, the shortcut formula can be used for other applications in probability theory and statistics, such as:

• Evaluating the joint distribution function of multiple random variables.
• Computing the probability of events involving multiple random variables.
• Deriving the distribution of functions of random variables.

Q: Are there any software packages or libraries that implement the shortcut formula?

A: Yes, there are several software packages and libraries that implement the shortcut formula, including:

• R: The integrate function in R can be used to evaluate the double-integral using the shortcut formula.
• Python: The scipy.integrate module in Python can be used to evaluate the double-integral using the shortcut formula.
• MATLAB: The int function in MATLAB can be used to evaluate the double-integral using the shortcut formula.