1.        The joint pdf of a two dimensional random variable (𝑋, 𝑌) is given by

         𝑓(𝑥, 𝑦) =

(i) Compute (a) 𝑃(X > 1) and (b)  𝑃 (Y < 5/2 )

(ii) Compute 𝑃 (X > 1/ Y < 5/2 )

 (iii) 𝑃(X + Y ≤ 3)   

To solve the given problems, we need to know the exact form of the joint probability density function (pdf) \( f(x, y) \) of the two-dimensional random variable \( (X, Y) \). However, since the explicit form of \( f(x, y) \) is missing from the problem statement, I'll guide you through the general approach to solving each part.

### General Approach:

1. **Identify the given joint pdf \( f(x, y) \).**
   - The joint pdf \( f(x, y) \) should be defined over a certain range for \( x \) and \( y \). It should also satisfy the property:
     \[
     \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, y) \, dx \, dy = 1
     \]
   - Based on the given \( f(x, y) \), we can find the marginal pdfs \( f_X(x) \) and \( f_Y(y) \) by integrating out the other variable.

2. **(i) Compute \( P(X > 1) \) and \( P(Y < 5/2) \).**
   - \( P(X > 1) \) is obtained by integrating the joint pdf over the appropriate region:
     \[
     P(X > 1) = \int_{1}^{\infty} \left( \int_{-\infty}^{\infty} f(x, y) \, dy \right) dx
     \]
   - \( P(Y < 5/2) \) is similarly obtained by integrating over:
     \[
     P(Y < 5/2) = \int_{-\infty}^{5/2} \left( \int_{-\infty}^{\infty} f(x, y) \, dx \right) dy
     \]

3. **(ii) Compute \( P(X > 1 \mid Y < 5/2) \).**
   - This is the conditional probability:
     \[
     P(X > 1 \mid Y < 5/2) = \frac{P(X > 1 \text{ and } Y < 5/2)}{P(Y < 5/2)}
     \]
   - \( P(X > 1 \text{ and } Y < 5/2) \) is found by integrating the joint pdf over the region where \( x > 1 \) and \( y < 5/2 \):
     \[
     P(X > 1 \text{ and } Y < 5/2) = \int_{1}^{\infty} \int_{-\infty}^{5/2} f(x, y) \, dy \, dx
     \]

4. **(iii) Compute \( P(X + Y \leq 3) \).**
   - This probability is found by integrating over the region where \( x + y \leq 3 \):
     \[
     P(X + Y \leq 3) = \int_{-\infty}^{\infty} \int_{-\infty}^{3-y} f(x, y) \, dx \, dy
     \]
   - This region is determined by the condition \( x + y \leq 3 \).

### Summary:

Without the specific form of \( f(x, y) \), the exact calculations can't be completed. However, the above steps outline how you would approach each part of the problem once \( f(x, y) \) is provided. If you provide the joint pdf \( f(x, y) \), I can perform the specific calculations for you.

Would you like to provide the form of \( f(x, y) \), or do you have any other questions?

Here are 5 related questions to explore further:
1. How do you determine the marginal distribution from a joint pdf?
2. What are the conditions for two random variables to be independent based on their joint pdf?
3. How do you compute the expected value of a function of two random variables given their joint pdf?
4. How would you find the correlation coefficient between \( X \) and \( Y \) given their joint pdf?
5. How can you visualize the joint pdf of two random variables using a 3D plot?

**Tip:** When working with joint distributions, always ensure that the pdf is normalized (integrates to 1) and carefully define the limits of integration based on the problem's constraints.

Explore how to compute probabilities involving the joint probability distribution (X, Y). Learn about marginal and conditional probabilities, and apply the Law of Total Probability and Bayes' Theorem. Suitable for advanced undergraduate students and researchers.

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