If the probability density function of a random variable x is f(x) = $f(x)=\{\begin{array}{l}\frac{1}{2}\sin x;0\le x\le \pi \\ 0:\text{ otherwise }\end{array}$. then mode of the distribution is:

Concept:

Mode is the value that has the greatest frequency density in its immediate neighborhood.

Method to Calculate Mode from PDF:

(1) Find critical point of PDF [i.e f^'(x) = 0]

(2) Check whether at the critical point function f(x) becomes maximum or not.

(3) If f(x) is maximum at the critical point, then the critical point itself becomes the mode.

Calculation:

Given;

 $f(x)=\{\begin{array}{l}\frac{1}{2}\sin x;0\le x\le \pi \\ 0:\text{ otherwise }\end{array}$. 

Calculation of critical point:

    $f^{\prime }(x)=0$

$\Rightarrow \frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{2}\sin x)=0$

$\Rightarrow \frac{\cos x}{2}=0$

$\therefore x=\frac{n\pi }{2}$

For; n = 1

x = π/2

Check whether it is maximum of f(x) or not:  

    $f^{″}(x)=0$

$\Rightarrow \frac{\mathrm{d}}{\mathrm{d}x}(\frac{\cos x}{2})=0$

$\Rightarrow \frac{-\sin x}{2}=0$

As f'' (x) becomes negative at x = π/2 → at x = π/2 , f(x) is maximum

Hence, Mode will be π/2