The c.d.f. $\mathrm{F}(x)$ associated with p.d.f. $\mathrm{f}(x)$ is

The c.d.f. $\mathrm{F}(x)$ associated with p.d.f. $\mathrm{f}(x)$ is
  1. (A) $\quad \mathrm{F}(x)=4 x^3+3 x^4$
  2. $\mathrm{F}(x)=4 x^3-3 x^4$
  3. $\mathrm{F}(x)=-4 x^3-3 x^4$
  4. $\mathrm{F}(x)=-4 x^3+3 x^4$

Solution

$\begin{aligned} \mathrm{F}(x) & =\int_0^x 12 x^2(1-x) \mathrm{d} x \\ & =12 \int_0^x\left(x^2-x^3\right) \mathrm{d} x \\ & =12\left[\frac{x^3}{3}-\frac{x^4}{4}\right]_0^x \\ & =12\left[\frac{x^3}{3}-\frac{x^4}{4}\right] \\ & =4 x^3-3 x^4\end{aligned}$

Asked in: MHT CET 2023 (13 May Shift 2)

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