# For the p.d.f $f(x) = \left\{ \begin{array}{l l} cx(1-x)^{3}, & \quad \text{0$<$x$<$1}\\ 0 & \quad \text{elsewhere} \end{array} \right.$ Find $p(x<\large\frac{1}{2})$

Toolbox:
• The probability density function (continuous probability function $f(x)$ satisfies the following properties :
• (i) $P(a\leq x\leq b)=\int_a^b f(x) dx$
• (ii) $f(x)$ is non-negative for all real $x$
• (iii) $\int_{-\infty}^\infty f(x) dx=1$
• Also $P(x=a)=0$
• $P(a\leq x\leq b)=P(a\leq x\leq b)$=P(a < x < b)
Step 1:
$f(x)=\left\{\begin{array}{1 1}cx(1-x)^3,& 0< x<1\\0 ,&elsewhere\end{array}\right.$ is a probability density function.
$\therefore f(x) \geq 0$ for all x and $\int_{-\infty}^\infty f(x) dx=1$
Step 2:
$\int_{-\infty}^\infty f(x) dx=1$
$\int_0^1 cx(1-x)^3dx=1$
(since $f(x)=0$ elsewhere)
$\Rightarrow c\bigg[\large\frac{x(1-x)^4}{4}-\large\frac{1}{-4}\frac{(1-x)^5}{-5}\bigg]_0^1$$=1 \Rightarrow \large\frac{-c(1-x)^4}{4}\bigg[x+\large\frac{1-x}{5}\bigg]_0^1$$=1$
$\Rightarrow \large\frac{c}{20}$$=1 c=20 Step 3: P(x <\large\frac{1}{2})=\int_{-\infty}^{\Large\frac{1}{2}}$$f(x)dx$
$\qquad\qquad=\int_0^{\Large\frac{1}{2}}20x(1-x^3)dx$
(since $f(x)=0$ elsewhere)
$\qquad\qquad=20\bigg[\large\frac{x(1-x)^4}{-4}-\frac{1}{20}$$(1-x)^5\bigg]_0^{\Large\frac{1}{2}} \qquad\qquad=\large\frac{-20}{4}$$(1-x)^4\bigg[x+\large\frac{1-x}{5}\bigg]_0^{\Large\frac{1}{2}}$
$\qquad\qquad=-5\bigg[(\large\frac{1}{2})^4(\large\frac{1}{2}+\frac{1}{10})-1(0+\large\frac{1}{5})\bigg]$
$\qquad\qquad=-5\bigg[\large\frac{1}{16}\times \frac{6}{10}-\frac{1}{5}\bigg]$
$\qquad\qquad=-5\bigg[\large\frac{3}{80}-\frac{1}{5}\bigg]$
$\qquad\qquad=-5\bigg[\large\frac{13}{80}\bigg]$
$\qquad\qquad=\large\frac{13}{16}$