random variable $X^2+Y^2 >1$

Question

Consider two independent random variables $X$ and $Y$ having probability density functions uniform in the interval $[-1, 1]$. The probability that $X^{2}+Y^{2}>1$ is 

• A. $\pi/4$
• B. $1-\pi/4$
• C. $\pi/2 - 1$
• D. Probability that $X^{2}+Y^{2}<0.5$
• E. None of the above.

Answer 1

Area of square denotes the total probability, i. e, 1.
Area of circle denotes P(X ² + Y ²  ≤ 1)
Area of shaded region denotes the required probability, i.e, P(X ²+ Y ² >1)

Area of shaded region=Area of square -Area of Circle
= 4 - ⊼
= 4 ( 1 - ⊼/4)

If area of square corresponds to total probability, then
4 sq.unit=1
= 1 sq.unit=1/4
= 4(1-⊼/4) sq.unit=
=1- ⊼/4 
which is the required probability.

Answer 2

(4-Pi)/4....is the answer

Comments

4 - Pi is the area outside circle, you need to divide by total area that is 4 also.

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yeahhh man...thanks!