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

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