Suppose a circle is drawn with random radius R, that is uniformly distributed between 0 and a constant c. How to find the expected value of the area Y = pi R² of such a circle?
2026-04-05 23:46:10.1775432770
Finding the expected values of the area of a circle
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$$E[Y] = E[\pi R^2] = \pi E[R^2] = \pi \cdot \int_{0}^{c}t^2f_R{t} = \pi \cdot \int_{0}^{c}t^2\cdot \frac{1}{c} = \pi \cdot \int_{0}^{c}t^2\cdot \frac{1}{c} = \pi \frac{c^2}{3} $$
Used: Linearity of expected value, Linearity of integrals,Smooth transformation of continous Random variable