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Chapter 3: Q5SE (page 93)

Suppose that\({{\bf{X}}_{\bf{1}}}\)and\({{\bf{X}}_{\bf{2}}}\)are i.i.d. random variables, and that each has the uniform distribution on the interval[0,1]. Evaluate\({\bf{P}}\left( {{{\bf{X}}_{\bf{1}}}^{\bf{2}}{\bf{ + }}{{\bf{X}}_{\bf{2}}}^{\bf{2}} \le {\bf{1}}} \right)\)

Short Answer

Expert verified

\(P\left( {{X_1}^2 + {X_2}^2 \le 1} \right) = \frac{\pi }{4}\)

Step by step solution

01

Given information

\({X_1}\)and\({X_2}\)are i.i.d. random variables. Both the variables have a uniform distribution on the interval [0,1].

02

Calculate the probability

\({X_1}\)and\({X_2}\)have the uniform distribution over a square, which has area 1.

The area of the quarter circle is\(\frac{\pi }{4}\)

Hence,\(P\left( {{X_1}^2 + {X_2}^2 \le 1} \right) = \frac{\pi }{4}\).

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