๐ Understanding the Support Region for Transformed Discrete Bivariate Variables
In the realm of mathematics, particularly when dealing with probability and statistics, understanding the support region of transformed variables is crucial. This article provides a comprehensive guide to grasping the concept, its history, key principles, and real-world applications.
๐ History and Background
The concept of variable transformation and support regions evolved with the development of probability theory and statistical analysis. Early statisticians recognized the need to manipulate variables to simplify analysis or meet certain assumptions. The formalization of support regions came later, providing a rigorous framework for understanding the possible values of these transformed variables.
๐ Key Principles
- ๐ Definition of Support: The support of a discrete bivariate variable $(X, Y)$ is the set of all possible pairs $(x, y)$ that have non-zero probability. Mathematically, it's the set $\{(x, y) : P(X=x, Y=y) > 0\}$.
- ๐ Transformation: A transformation involves creating new variables $U$ and $V$ from the original variables $X$ and $Y$. This can be represented as $U = g(X, Y)$ and $V = h(X, Y)$, where $g$ and $h$ are functions.
- ๐ Determining the New Support: To find the support region for the transformed variables $(U, V)$, you need to determine the possible values of $U$ and $V$ based on the support of $(X, Y)$ and the transformation functions $g$ and $h$. This involves solving for $u$ and $v$ given all possible pairs of $(x, y)$.
- ๐บ๏ธ Mapping: Consider each point $(x, y)$ in the support of $(X, Y)$. Apply the transformations to find the corresponding point $(u, v)$, where $u = g(x, y)$ and $v = h(x, y)$. The set of all such $(u, v)$ forms the support of $(U, V)$.
- ๐ก Example: Let's say $X$ and $Y$ can each take values 0, 1, and 2. The transformation is $U = X + Y$ and $V = X - Y$. We need to find all possible values of $U$ and $V$ given the possible pairs of $(X, Y)$.
โ๏ธ Real-world Examples
Consider a scenario where $X$ represents the number of successful sales calls a person makes in a day, and $Y$ represents the number of follow-up emails sent. Both $X$ and $Y$ are discrete variables. A transformation might involve defining $U = X + Y$ (total activity) and $V = X / Y$ (success rate per email). The support region of $(U, V)$ would then provide insights into the range of total activity and success rates achievable.
๐งฎ Example with Calculations
Suppose $X$ and $Y$ are independent Bernoulli random variables with $P(X=1) = p$ and $P(Y=1) = q$. Let's define $U = X + Y$ and $V = X - Y$. The support of $(X, Y)$ is $\{(0, 0), (0, 1), (1, 0), (1, 1)\}$.
Now, we find the support of $(U, V)$:
- ๐ For $(X, Y) = (0, 0)$, $U = 0 + 0 = 0$ and $V = 0 - 0 = 0$. So, $(U, V) = (0, 0)$.
- ๐ For $(X, Y) = (0, 1)$, $U = 0 + 1 = 1$ and $V = 0 - 1 = -1$. So, $(U, V) = (1, -1)$.
- ๐ For $(X, Y) = (1, 0)$, $U = 1 + 0 = 1$ and $V = 1 - 0 = 1$. So, $(U, V) = (1, 1)$.
- ๐ For $(X, Y) = (1, 1)$, $U = 1 + 1 = 2$ and $V = 1 - 1 = 0$. So, $(U, V) = (2, 0)$.
Therefore, the support of $(U, V)$ is $\{(0, 0), (1, -1), (1, 1), (2, 0)\}$.
๐ Conclusion
Understanding the support region for transformed discrete bivariate variables is vital for various statistical analyses. By systematically mapping the original support to the new variables, we can accurately determine the possible values and probabilities associated with the transformed variables. This knowledge is crucial for making informed decisions and drawing meaningful conclusions from data.