91Ó°ÊÓ

In probability theory, a normalization constant ensures that a probability density function (PDF) correctly represents a valid probability distribution. For a function to be a PDF, the total probability across its entire range must equal 1. This is where the normalization constant, often represented as \( k \), comes into play.
To determine \( k \), we integrate the given joint probability density function (PDF) over its full range. For our given function \( f(y_1, y_2) = k y_1 y_2 \), where \( 0 \leq y_1, y_2 \leq 1 \), we find:
\[ \int_0^1 \int_0^1 k y_1 y_2 \, dy_1 \, dy_2 = 1. \]
This is because the integral of a probability density function (PDF) over its entire space must equal to 1. By solving this double integral, we discovered that \( k = 4 \).
This normalization constant adjusts the function ensuring the area under the curve equals 1, thus confirming it as a valid probability density function. So, without \( k \), we wouldn't correctly represent the distribution!

• The total area under the curve of a PDF must always equal 1.
• \( k \) ensures the function is properly scaled to meet this criterion.

The joint distribution function is a critical concept in understanding how two random variables are related. Unlike a single variable, the joint distribution considers the likelihood of both variables occurring together within a specified range.
In our case, we seek the joint distribution function \( F(y_1, y_2) \). This function is derived by integrating the joint probability density function (PDF) with respect to both variables.
The formula to achieve this is:
\[ F(y_1, y_2) = \int_0^{y_2} \int_0^{y_1} 4 y_1' y_2' \, dy_1' \, dy_2', \]
where \( y_1' \) and \( y_2' \) are dummy variables meant for integration simplicity. After performing the integration, we find:
\[ F(y_1, y_2) = y_1^2 y_2^2. \]

• This joint distribution function now tells us the cumulative probability for events described by \( y_1 \) and \( y_2 \) together.
• It enables probability calculations over a specific range of these two variables.

Understanding this joint distribution function helps in grasping how different values of \( y_1 \) and \( y_2 \) interact and the probability of their simultaneous occurrences.

Calculating probabilities for specific events using the joint distribution function is a practical application of the theoretical concepts we've learned.
Suppose we want to find the probability that \( Y_1 \leq 1/2 \) and \( Y_2 \leq 3/4 \). We simply substitute these values into our joint distribution function \( F(y_1, y_2) = y_1^2 y_2^2 \).
Here's the computation:
\[ F\left( \frac{1}{2}, \frac{3}{4} \right) = \left( \frac{1}{2} \right)^2 \left( \frac{3}{4} \right)^2. \]
This equals:
\[ = \frac{1}{4} \cdot \frac{9}{16} = \frac{9}{64}. \]
The result, \( \frac{9}{64} \), is the probability that \( Y_1 \) and \( Y_2 \) fall within those specified limits.
Such computations are invaluable when determining the likelihood of various outcomes within a joint probability framework.

• Using the joint distribution function allows for precise probability calculations for specific conditions.
• Simplifying these probabilistic scenarios supports data-driven decision-making and predictions.