Suppose that $X$ and $Y$ are iid uniform distribution with $U(0, 1)$ random variables.
(a) What is $\mathbb P((X, Y ) ∈ [a, b]×[c, d])$ for $0 ≤ a ≤ b ≤ 1$ and $0 ≤ c ≤ d ≤ 1$ ?
What is $\mathbb P((X, Y ) ∈ A)$, where $A$ is an arbitrary subset of $[0, 1] × [0, 1]$?
(b) Let $A = {(x, y) ∈ [0, 1] × [0, 1] : x^2 + y^2 ≤ 1}$. What is the area of $A$ ?
(c). Define the rv $Z$ by
$$Z = 1 \quad ,\text{if} X^2 + Y^2 ≤ 1,$$ $$=0\quad, \text{otherwise}$$ What is $\mathbb E[Z]$?
(d) By simulating $Z$, write a program to estimate $π$.
My attempt:
(a)$\mathbb P((X, Y ) ∈ [a, b]×[c, d])=\mathbb P(X)\mathbb P(Y)$
since $X$ and $Y$ are independent
$\mathbb P((X, Y ) ∈ [a, b]×[c, d])=\mathbb P(X)\mathbb P(Y)=(1)(1)=1$
for $0 ≤ a ≤ b ≤ 1$ and $0 ≤ c ≤ d ≤ 1$
I don't know what will be $\mathbb P((X, Y ) ∈ A)$ and (b)
for (c) i can write R code but don't know to manipulate at hand.
# R code
n <- 100 # I choose n=100 randomly
X <- runif(n,0,1)
Y <- runif(n,0,1)
Z <- (X^2+Y^2<=1)
EZ <- sum(Z)/n
(d) ???
a) As @Macavity says $P((X,Y)∈[a,b]×[c,d])=P(X∈[a,b])⋅P(Y∈[c,d])$, since $X$ and $Y$ are $U(0,1)$, $P(X∈[a,b])=b-a$ and $P(Y∈[c,d])=d-c$ so $P((X,Y)∈[a,b]×[c,d])=(b-a)(d-c)$
Because $X$ and $Y$ are $U(0,1)$ the combined distribution $(X,Y)$ is uniformly distributed over the unit square with corners at $(0,0)$ and $(1,1)$. For any arbitrary subset $A$ of this square $P(X,Y)\in A$ is equal to the area of $A$.
b) For $A=(x,y)\in [0,1]×[0,1]:x^2+y^2\le 1$,
$$\begin{align} x^2+y^2&\le1\\ y^2&\le1-x^2\\ y&\le\sqrt{1-x^2}, \text{since all numbers are non-negative}\\ \end{align} $$
Integration will give you the area.
c) The area you just worked out will give you the probability $p$ that $Z$ is $1$. $E(Z)=p\times 1+(1-p)\times 0=p$
d) What is the shape of $X^2+Y^2\le 1$ within the unit square? How does the area of that shape relate to $\pi$? If you simulate $Z$ enough what value does the mean of your simulations approach? So, how does this relate to $\pi$?