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Let $ f(x) = 30x^2 (1 - x)^2 $ for $ 0 \le x \le 1 $ and $ f(x) = 0 $ for all other values of $ x $.

(a) Verify that $ f $ is a probability density function.

(b) Find $ P(X \le \frac{1}{3}) $.

(A) $$\left[10 x^{3}-15 x^{4}+6 x^{5}\right]_{0}^{1}=10-15+6=1$$

(B) $$\frac{10}{27}-\frac{15}{81}+\frac{6}{243}=\frac{17}{81}$$

Applications of Integration

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