Problem 134 - An Introduction to Mathematical Statistics and Its Applications

Question

Find \(c\) if \(f_{X, Y}(x, y)=c x y\) for \(X\) and \(Y\) defined over the triangle whose vertices are the points \((0,0),(0,1)\), and \((1,1)\).

Step-by-Step Solution

Answer

The constant \(c\) for the joint probability density function \(f_{X, Y}(x, y)\) is 8.

Step 1: Set Up the Double Integral

Since we know that over the entire support, the probability must sum to 1, we can set up a double integral. We know that we are integrating over a triangle with vertices at \((0,0),(0,1)\), and \((1,1)\), can use these points to set up our double integral.\[\int_0^1 \int_0^x cxy\,dy\,dx = 1\]

Step 2: Evaluation of the Inner Integral

Keeping \(x\) as constant, integrate the inner integral with respect to \(y\). We get, \[\int_0^x cxy\,dy = cx \int_0^x y\,dy = cx[\frac{y^2}{2}|_0^x =\frac{c x^3}{2}\]

Step 3: Evaluation of the Outer Integral

Now, integrate the result from Step 2 with respect to x between 0 and 1. \[\int_0^1 \frac{c x^3}{2}\,dx = \frac{c}{2} \int_0^1 x^3\,dx = \frac{c}{2}[\frac{x^4}{4}|_0^1 =\frac{c}{8}\] This should equal 1.

Step 4: Solve for the Constant \(c\)

Solving the equation \(\frac{c}{8} = 1\) for \(c\), we find that \(c = 8\).

Key Concepts

Double Integral

The concept of a double integral extends the idea of an integral to functions of two variables. Consider a function f(x, y) over a certain region in the xy-plane. The double integral of f over this region gives the volume under the surface defined by f above the xy-plane and above the region of integration.

For instance, if you wanted to integrate over a rectangle, you’d specify limits for both x and y that define this rectangle. But when dealing with a triangle or any other non-rectangular shape, setting the limits requires more attention, often involving the equations that constitute the edges of the region.

An example double integral would look like this:\[\int_{a}^{b} \int_{c(x)}^{d(x)} f(x,y)\,dy\,dx\]Here, the inner integral is first evaluated over y, holding x constant, and then the outer integral is evaluated over x. Such calculations are fundamental in calculus, especially when assessing probabilities for continuous random variables.

Integration Bounds

The term integration bounds refers to the limits between which integration is performed. In the context of a double integral, there are two sets of bounds: one for each variable of integration (x and y). The bounds for y can depend on x, which is often the case when integrating over a non-rectangular shape such as a triangular region.

In the example problem, the bounds for y are from 0 to x, which implies that for each value of x, y takes on values starting from 0 up to the value of x. This corresponds to one side of the triangle being along the line y = x. The bounds for x are from 0 to 1, which are constants and define the range of x over which we integrate. The correct choice and understanding of integration bounds are crucial for accurately evaluating a double integral.

Probability Normalization

The principle of probability normalization states that the total probability across the entire probability space must equal 1. This is true for discrete random variables, where the sum of the probabilities equals 1, and for continuous random variables, where the integral of the probability density function (pdf) over the entire space equals 1.

In the context of our exercise, the double integral of the joint pdf f(x, y) over its domain (the triangular region) must equal 1. This is a form of normalization and it ensures that the distribution is valid and properly defines the probabilities over the space where the two variables X and Y live. Solving the double integral with the condition that it equals 1 allows us to determine the constant c, which is necessary to normalize the distribution.

Continuous Random Variables

Continuous random variables are quantities that can take on an infinite number of values within a given range. Unlike discrete variables, which have distinct and separate values, continuous variables can represent any value within a range. The probabilities associated with continuous random variables are described by probability density functions (pdfs), not probability mass functions.

For continuous random variables, the probability of any single, exact value is essentially zero; instead, probabilities are assigned to intervals. These probabilities are calculated using integrals of the pdf over the specified ranges. The exercise deals with a joint pdf, f(x, y), representing two continuous random variables and their combined behavior. To ensure that their combined pdf is correctly defined, it must be normalized by finding the right constant c, as done in the exercise, so that the total probability across the entire range is 1.