I have two continuous random variables $X$ and $Y$ distributed according to the joint density function $f_{X, Y}(x, y)=K(x+y)$ for $0 \leq x, y \leq 1$, and 0 otherwise.
I am tasked with calculating the conditional probability $P(X>0.5 \mid X=Y)$. Here's the exact setup:
- Joint Density Function: $$ f_{X, Y}(x, y)=K(x+y), \quad 0 \leq x, y \leq 1 $$
The constant $K$ is determined by the condition that the total probability over the region must be 1: $$ \int_0^1 \int_0^1 K(x+y) d x d y=1 $$ 2. Conditional Probability: I need to compute $P(X>0.5 \mid X=Y)$. However, since $X=Y$ represents a line in the continuous space, I suspect that $P(X=Y)=0$, and the standard formula for conditional probability: $$ P(X>0.5 \mid X=Y)=\frac{P(X>0.5 \text { and } X=Y)}{P(X=Y)} $$ cannot be directly applied because $P(X=Y)=0$.
My Questions/Concerns:
- Is $P(X=Y)=0$ correct? Since we're dealing with continuous random variables, does this imply that $P(X=Y)=0$, making the standard conditional probability formula inapplicable?
- How can I calculate $P(X>0.5 \mid X=Y)$ in this case? Should I use a conditional density approach or limit arguments? How should I approach this when $P(X=Y)$ is zero?
- Finding the constant $K$ : I have already computed that $K=1$ by solving: $$ \int_0^1 \int_0^1 K(x+y) d x d y=1 $$ yielding $K=1$. Does this affect how I should proceed with the conditional probability calculation?
Attempted Solution:
- After finding $K=1$, I considered the possibility that along the line $X=Y$, the marginal density would be uniform since the joint density $f_{X, Y}(x, y)=x+y$ is symmetric. Hence, the probability $P(X>0.5 \mid X=Y)$ might be 0.5 , similar to a uniform distribution on $[0,1]$
- However, I am not entirely sure how to formalize this intuition. Should I approximate the conditional probability by looking at neighbourhood around $X=Y$, or is there a more rigorous method using conditional densities?
Any guidance or clarification would be greatly appreciated!
