Problem 151 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose that \(f_{X, Y}(x, y)=6(1-x-y)\) for \(x\) and \(y\) defined over the unit square, subject to the restriction that \(0 \leq x+y \leq 1\). Find the marginal pdf for \(X\).

Step-by-Step Solution

Answer

The marginal pdf for \(X\) can be found by integrating the joint pdf over all possible values of \(Y\), considering the given limits. Complete the integration, and simplify the obtained expression to get the final answer.

Step 1: Preparing to Integrate

Firstly, you need to set up the integral. Recall that the marginal pdf of \(X\), noted \(f_X(x)\), is obtained by integrating the joint pdf \(f_{X, Y}(x, y)\) over all possible values of \(Y\). The interdependence of the variables \(X\) and \(Y\) determines the limits of integration. According to the given condition \(x + y \leq 1\), the limits of \(Y\) are \(0\) to \(1-x\).

Step 2: Setting up the Integral

So now, the marginal pdf of \(X\) can be set up as:\[f_X(x) = \int_0^{1-x} f_{X,Y}(x,y) \, dy\]Substitute the given values of joint pdf \(f_{X,Y}(x,y) = 6(1-x-y)\)

Step 3: Perform the Integration

Carry out the integral over \(y\):\[f_X(x) = \int_0^{1-x} 6(1 - x - y) \, dy\]This is a simple standard integration task, where you need to integrate \(y\) and subtract the results at the given limits.

Step 4: Simplify the Expression

After calculating the integral, simplify the resulting expression to form a closed equation that represents the marginal pdf of \(X\).

Key Concepts

Joint Probability Density Function

In probability theory, the joint probability density function (joint pdf) gives us the probability distribution for two continuous random variables, say \(X\) and \(Y\). It describes how likely different pairs of values for \(X\) and \(Y\) are to occur.

For instance, in the exercise provided, we have the joint pdf \(f_{X, Y}(x, y) = 6(1-x-y)\). This function defines the probability density over the unit square region defined by the condition \(0 \leq x+y \leq 1\).

Whenever dealing with joint pdfs:

• The integral of the joint pdf over the entire distribution must equal 1, ensuring that it corresponds to a valid probability distribution.
• The limits of integration can vary depending on the restrictions placed on \(x\) and \(y\). These are crucial in applying joint pdfs.
• Understanding the relationship between \(X\) and \(Y\) is essential for calculating marginal probabilities.

Integration in Probability

Integration is essentially used in probability when dealing with continuous random variables. It helps us to accumulate probabilities over a range of values, a concept similar to summing probabilities from discrete distributions.

In our exercise, we use integration to find the marginal probability density function for \(X\), noted as \(f_X(x)\). This means we integrate the joint pdf over the variable \(Y\). Here's how it works:

• We begin the integral from the lower limit of \(0\) to the upper limit of \(1-x\), due to the condition \(x + y \leq 1\).
• Substitute the provided joint pdf \(f_{X,Y}(x,y) = 6(1-x-y)\) into the integral equation \( \ f_X(x) = \int_0^{1-x} 6(1-x-y) \, dy \)
• After performing the integration, remember to simplify any resulting expressions to achieve a more interpretable form of the marginal pdf.

This process allows us to see how likely different values of \(X\) are when considering all possible \(Y\) values in the distribution.

Mathematical Statistics

Mathematical statistics involves the application of mathematics to analyze random phenomena. It requires using concepts like joint and marginal probability density functions. Knowing these is crucial for understanding the behavior of random variables within a given dataset.

From our exercise, the steps involve combining both algebraic manipulation and integration techniques to extract meaningful statistics from a joint distribution:

• Identifying the boundary conditions (here, \(0 \leq x+y \leq 1\)) to establish correct limits for integration.
• Performing definite integration to transition from joint to marginal probability distributions.
• Ability to reduce and simplify expressions helps convey the marginal distribution in a clear form.

Overall, mathematical statistics provide the foundational framework needed across various scientific disciplines to analyze and interpret data, ensuring informed decision-making based on statistical inference.