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) ?

We are given the joint density function \( f(x, y) \) of the random variables \( X \) and \( Y \):

\[
f(x, y) = 
\begin{cases} 
e^{-(x + y)} & \text{for } 0 \leq x, y < \infty, \\
0 & \text{otherwise}.
\end{cases}
\]

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.

Let X and Y have the joint density f(x, y) = e^(-(x+y)) for 0 ≤ x, y < ∞, 0 otherwise. What is P(X ≥ Y ≥ 2)?

This problem explores how to find the probability P(X ≥ Y ≥ 2) for random variables X and Y with a joint density f(x, y) = e^{-(x + y)}. The solution uses double integrals and properties of exponential distributions to compute the exact probability. Suitable for college-level students studying probability and statistics.

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