Probability , Geometric and Gaussian

Question

So,I'm good at the questions which require the understanding of basic formulaes , but this one my prof said needs me to think (for the first one)'geometrically'=Stumped. Please Help! The second is an extra credit and I'm having trouble understanding the problem itself , attempting at solving is far fetched :(

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Answer 1

Since the second question is for extra credit, I feel I can explain the meaning of any terms you find unfamiliar, and no more. You have not yet specified what it is about the questions that you do not understand.

For the first question, the pair $(X,Y)$ is uniformly distributed in the square with corners $(0,0)$, $(1,0)$, $(1,1)$, and $(0,1)$. Draw that square. We are given that $X+Y\le 1$. So $(X,Y)$ lies in the square and below the line $x+y=1$. Draw that line.

Given that $(X,Y)$ lies below the line, we want to find various probabilities. Let $T$ be the triangle with corners $(0,0)$, $(1,0)$, and $(1,1)$. This has area $1/2$.

(a) $\Pr(|X-1|\lt 1)$ is very easy, conditional or not. For with probability $1$, this happens.

(b) This one is the hardest. Draw the curve $xy=1$. Find the area $b$ of the part of $T$ that lies below that line, and divide by the area of $T$. To calculate the area $b$, you will need to integrate.

(c) Draw the lines $x=1/2$ and $y=1/2$. Let $c$ be the area of the part of $T$ that lies to the left of $x=1/2$ and below $y=1/2$. Then divide by the area of $T$.

(d) Draw the circle $x^2+y^2=1/4$. Find the area $d$ of the part of the disk that lies inside $T$, and divide by the area of $T$.