Step 1: Concept Overview:
Given the joint cumulative distribution function (CDF) of random variables X and Y, we aim to determine the variance of X. This involves deriving the marginal CDF of X, followed by its marginal probability density function (PDF). The PDF allows us to either identify the distribution or directly compute the variance.
Step 2: Core Formulas:
1. Determine the marginal CDF of X, \(F_X(x)\), by evaluating the limit of the joint CDF as \(y\) approaches infinity:
\[ F_X(x) = \lim_{y \to \infty} F_{X,Y}(x,y) \]
2. Compute the marginal PDF of X, \(f_X(x)\), by differentiating the marginal CDF:
\[ f_X(x) = \frac{d}{dx} F_X(x) \]
3. Identify the distribution from its PDF. A PDF of the form \(f(x) = \lambda e^{-\lambda x}\) for \(x>0\) indicates an exponential distribution with rate \(\lambda\).
4. The variance of an exponential distribution is given by: \(\text{Var}(X) = \frac{1}{\lambda^2}\).
Step 3: Step-by-Step Solution:
First, calculate the marginal CDF of X for \(x>0\):
\[ F_X(x) = \lim_{y \to \infty} (1 - e^{-x} - e^{-y} + e^{-(x+y)}) \]
As \(y\) approaches infinity, \(e^{-y}\) approaches 0, and \(e^{-(x+y)} = e^{-x}e^{-y}\) also approaches 0.
\[ F_X(x) = 1 - e^{-x} - 0 + 0 = 1 - e^{-x} \]
Thus, the marginal CDF of X is \(F_X(x) = 1 - e^{-x}\) for \(x>0\).
Next, find the marginal PDF of X by differentiating \(F_X(x)\):
\[ f_X(x) = \frac{d}{dx}(1 - e^{-x}) = -(-e^{-x}) = e^{-x} \]
The PDF is \(f_X(x) = e^{-x}\) for \(x>0\), which is the PDF of an exponential distribution with rate parameter \(\lambda = 1\).
Finally, calculate the variance of X. For an exponential distribution with rate \(\lambda\), the variance is \(1/\lambda^2\).
\[ \text{Var}(X) = \frac{1}{1^2} = 1 \]
Alternatively, observe the factorization of the joint CDF:
\[ F_{X,Y}(x,y) = 1 - e^{-x} - e^{-y} + e^{-x}e^{-y} = (1 - e^{-x})(1 - e^{-y}) = F_X(x)F_Y(y) \]
This confirms that X and Y are independent and both follow an exponential distribution with \(\lambda = 1\).
Step 4: Answer:
The variance of X is 1.