91Ó°ÊÓ

If the chance that the random vector lies in CR, we say that (X, Y) is uniformly distributed throughout a region R of finite area A(R).

Let R²be the two - dimensional plane.

Then observe that for $E⊆R^2$

$P\left\{\right(X,Y)∈E\}={âˆ\neg }_{E}f(x,y)dxdy={âˆ\neg }_{Eâˆ\copyright R}c·area(Eâˆ\copyright R)$

Since $f(x,y)$is a joint density function,

We have

$1=P\left\{(X,Y)∈R^2\right\}=c·area\left(R^2âˆ\copyright R\right)=c·area\left(R\right)$

Therefore,

The area of $Ris1/c$.

Let R be the real line.

For sets$A,B⊆R$

We have

$P(X∈A,Y∈B)={∫}_B{∫}_{A}f(x,y)dxdy={∫}_{Bâˆ\copyright âˆ\pounds -1,1]}{∫}_{Aâˆ\copyright âˆ\pounds -1,1]}\frac{1}{4}dxdy$

=$\frac{1}{4}\text{length}(Aâˆ\copyright [-1,1\left]\right)\text{length}(Bâˆ\copyright [-1,1\left]\right)$

And

$P\{X∈A\}={∫}_{-2}^{*}{∫}_{A}f(x,y)dxdy={∫}_{-1}^1{∫}_{AN-1,1]}\frac{1}{4}dxdy$

$=\frac{1}{2}\text{length}(Aâˆ\copyright [-1,1\left]\right)$

And

$P\{Y∈B\}={∫}_{-∞}^{≈}{∫}_{B}f(x,y)dxdy={∫}_{-1}^1{∫}_{BN-1,1]}\frac{1}{4}dxdy$

$=\frac{1}{2}\text{length}(Bâˆ\copyright [-1,1\left]\right)$

Thus,

$P\{X∈A,Y∈B\}=\frac{1}{4}\text{length}(Aâˆ\copyright [-1,1\left]\right)\text{length}(Bâˆ\copyright [-1,1\left]\right)$

$=P\{X∈A\}P\{Y∈B\}$

Therefore,

X and Y are independent.

Also,

The above formulas for$P\{X∈A\}andP\{Y∈B\}$show that each of $XandY$are uniformly distributed over $[-1,1]$.

Let C be the interior of the circle of radius 1 centered at the origin.

Then we are required to compute $P\left\{\right(X,Y)∈C\}$.

Thus,

$P\left\{\right(X,Y)∈C\}=\frac{1}{4}·area(Câˆ\copyright R)=\frac{1}{4}·area\left(C\right)=\frac{1}{4}·\mathrm{Ï€}=\frac{\mathrm{Ï€}}{4}$