Marginal distribution of the unit disk

Question

can you help me how to solve this task, I need also an explenation on formal definitions how to proceed with that question

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Answer 1

a)

1)$\ X$ is uniformly distributed, hence $f_X(x_1, x_2) = \frac{1}{\pi}, \ x_1^2 + x_2^2 \le 1$

2) Marginal distribution of $X_1$ is: $$f_{X_1}(x_1) = \int f_X(x_1, x_2)dx_2 = \int_{-\sqrt{1-x_1^2}}^{\sqrt{1-x_1^2}} \frac{1}{\pi}dx_2 = \frac{2\sqrt{1 - x_1^2}}{\pi}, \ x_1 \in [-1, 1]$$

3)Obviously $X_1, X_2$ are dependent, because $f_X(x_1, x_2) \ne f_{X_1}(x_1)f_{X_2}(x_2)$

b)

1)The event $\sqrt{X_1^2 + X_2^2} \le r$ is the same as the event: $X_1$ and $X_2$ are inside the disc of the radius $r$. The probability of such an event is $\frac{\pi r^2}{\pi} = r^2$. So, the CDF$$F_{\sqrt{X_1^2 +X_2^2}}(r) = r^2$$