joint DF is given by FXY (x,y) = {(1-e^-ax) (1-e^-by) x,y>=0 and a,b>=0 and  if 0 else 

Determine if X and Y are dependent or independent. Also, what are P(X<=x, Y<Y)

2) P(X>x, Y>y)

Given the joint distribution function \( F_{XY}(x, y) \) as:

\[
F_{XY}(x, y) = 
\begin{cases}
(1 - e^{-ax})(1 - e^{-by}) & \text{if } x, y \geq 0 \text{ and } a, b \geq 0, \\
0 & \text{otherwise}.
\end{cases}
\]

### 1. Determine if \( X \) and \( Y \) are independent or dependent:

Two random variables \( X \) and \( Y \) are independent if the joint cumulative distribution function (CDF) can be written as a product of the marginal CDFs:

\[
F_{XY}(x, y) = F_X(x) \cdot F_Y(y).
\]

From the given CDF:

\[
F_{XY}(x, y) = (1 - e^{-ax})(1 - e^{-by}).
\]

Now, let’s check the marginal CDFs:

- The marginal CDF of \( X \), \( F_X(x) \), is obtained by setting \( y \to \infty \) (since \( F_{XY}(x, \infty) = F_X(x) \)):
  \[
  F_X(x) = 1 - e^{-ax}.
  \]

- Similarly, the marginal CDF of \( Y \), \( F_Y(y) \), is obtained by setting \( x \to \infty \):
  \[
  F_Y(y) = 1 - e^{-by}.
  \]

Thus, we can see that the joint CDF \( F_{XY}(x, y) \) is indeed the product of the marginal CDFs:

\[
F_{XY}(x, y) = F_X(x) \cdot F_Y(y).
\]

Therefore, \( X \) and \( Y \) are **independent**.

---

### 2. \( P(X \leq x, Y \leq y) \):

By definition, this is the joint CDF:

\[
P(X \leq x, Y \leq y) = F_{XY}(x, y) = (1 - e^{-ax})(1 - e^{-by}).
\]

---

### 3. \( P(X > x, Y > y) \):

This probability can be computed as:

\[
P(X > x, Y > y) = 1 - P(X \leq x \text{ or } Y \leq y).
\]

Since \( X \) and \( Y \) are independent, we can express this as:

\[
P(X > x, Y > y) = P(X > x) \cdot P(Y > y).
\]

From the marginal CDFs, we know:

- \( P(X > x) = 1 - F_X(x) = e^{-ax} \),
- \( P(Y > y) = 1 - F_Y(y) = e^{-by} \).

Thus,

\[
P(X > x, Y > y) = e^{-ax} \cdot e^{-by} = e^{-ax - by}.
\]

---

### Summary:

1. \( X \) and \( Y \) are **independent** because their joint CDF is the product of their marginal CDFs.
2. \( P(X \leq x, Y \leq y) = (1 - e^{-ax})(1 - e^{-by}) \).
3. \( P(X > x, Y > y) = e^{-ax - by} \).

---

Would you like further details or any clarification?

Here are 5 questions related to this:

1. How can you derive the marginal probability density functions (PDFs) for \( X \) and \( Y \)?
2. What is the joint probability density function (PDF) of \( X \) and \( Y \)?
3. How would the solution change if \( X \) and \( Y \) were dependent?
4. Can we compute \( P(X > x, Y \leq y) \)?
5. How does the value of parameters \( a \) and \( b \) influence the probabilities?

**Tip**: For independent random variables, you can always express joint probabilities as the product of marginal probabilities.

The query submitted by the user: joint DF is given by FXY (x,y) = {(1-e^-ax) (1-e^-by) x,y>=0 and a,b>=0 and if 0 else. Determine if X and Y are dependent or independent. Also, what are P(X<=x, Y<=y) and P(X>x, Y>y)?

This problem determines whether two random variables X and Y are independent based on their joint distribution function F_{XY}(x, y) = (1 - e^{-ax})(1 - e^{-by}). Learn how to compute the probabilities P(X<=x, Y<=y) and P(X>x, Y>y) using key formulas in probability theory. Suitable for undergraduate students studying probability and statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation