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.