Math Problem Statement
Let X and Y have the joint density f(x, y) = e^(−(x+y)) f(or 0 ≤ x, y < ∞ 0 otherwise. What is P (X ≥ Y ≥ 2) ?
Solution
We are given the joint density function of the random variables and :
\begin{cases} e^{-(x + y)} & \text{for } 0 \leq x, y < \infty, \\ 0 & \text{otherwise}. \end{cases}$$ We are tasked with finding $$P(X \geq Y \geq 2)$$. ### Step 1: Defining the Probability The probability $$P(X \geq Y \geq 2)$$ translates to the following region in the $$xy$$-plane: $$P(X \geq Y \geq 2) = P( X \geq 2 \text{ and } Y \geq 2 \text{ and } X \geq Y ).$$ This means we want to integrate the joint density function over the region defined by: - $$Y \geq 2$$, - $$X \geq Y$$. ### Step 2: Setting Up the Integral The region of integration is: - $$y \geq 2$$, - For each fixed $$y$$, $$x \geq y$$. The probability is given by the double integral: $$P(X \geq Y \geq 2) = \int_{2}^{\infty} \int_{y}^{\infty} e^{-(x + y)} \, dx \, dy.$$ ### Step 3: Evaluating the Inner Integral First, evaluate the inner integral with respect to $$x$$, while treating $$y$$ as constant: $$\int_{y}^{\infty} e^{-(x + y)} \, dx = e^{-y} \int_{y}^{\infty} e^{-x} \, dx = e^{-y} \left[ -e^{-x} \right]_{x=y}^{x=\infty}.$$ At the bounds, this becomes: $$e^{-y} \left( 0 - (-e^{-y}) \right) = e^{-y} \cdot e^{-y} = e^{-2y}.$$ ### Step 4: Evaluating the Outer Integral Now, evaluate the outer integral: $$\int_{2}^{\infty} e^{-2y} \, dy.$$ This is a simple exponential integral: $$\int_{2}^{\infty} e^{-2y} \, dy = \frac{1}{2} \left[ -e^{-2y} \right]_{2}^{\infty} = \frac{1}{2} \left( 0 - (-e^{-4}) \right) = \frac{1}{2} e^{-4}.$$ ### Step 5: Conclusion Thus, the probability $$P(X \geq Y \geq 2)$$ is: $$P(X \geq Y \geq 2) = \frac{1}{2} e^{-4}.$$ ### Final Answer: $$P(X \geq Y \geq 2) = \frac{1}{2} e^{-4}.$$ --- Would you like a deeper explanation on any of these steps? Here are some related questions to explore further: 1. What is the marginal distribution of $$X$$ and $$Y$$ given their joint density? 2. How would you compute $$P(X + Y \geq 4)$$ for the same joint density? 3. Can you find the expected value $$E[X]$$ given this joint density function? 4. How would you approach $$P(X \geq 1, Y \geq 1)$$ using this density function? 5. What is the covariance of $$X$$ and $$Y$$ for this joint distribution? **Tip:** In problems involving joint densities, sketching the region of integration can help clarify the limits for double integrals.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability
Joint Probability Density Function
Double Integrals
Exponential Distribution
Formulas
Joint density function: f(x, y) = e^{-(x+y)} for 0 ≤ x, y < ∞
P(X ≥ Y ≥ 2) = ∫∫ e^{-(x + y)} dx dy over the specified region
Integral evaluation: ∫_y^∞ e^{-(x+y)} dx and ∫_2^∞ e^{-2y} dy
Theorems
Properties of Exponential Distribution
Fubini’s Theorem for double integrals
Suitable Grade Level
College level (Probability and Statistics)
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