Concept:

Probability Density Function (PDF): It is the probability function which is represented for the density of a continuous random variable lying between a certain range of values. If x is the continuous random variable with density function f(x), then for a valid probability density function the area between the density curve and horizontal X-axis must be equal to 1. It means

\(\mathop \smallint \limits_{ - \infty }^\infty {\rm{f}}\left( {\rm{x}} \right){\rm{dx}} = 1\) 

Calculation:

\(\begin{array}{*{20}{c}}{{\rm{Given}},{\rm{\;\;f}}\left( {\rm{x}} \right) = {\rm{\lambda }}\left( {{\rm{x}} - 1} \right)\left( {2 - {\rm{x}}} \right)}&{{\rm{for\;}}1 \le {\rm{x}} \le 2}\\{ = 0}&{{\rm{otherwise}}}\end{array}\) 

\(\therefore {\rm{\;}}\mathop \smallint \limits_{ - \infty }^\infty {\rm{f}}\left( {\rm{x}} \right){\rm{dx}} = \mathop \smallint \limits_1^2 {\rm{f}}\left( {\rm{x}} \right){\rm{dx}} = \mathop \smallint \limits_1^2 {\rm{\lambda }}\left( {{\rm{x}} - 1} \right)\left( {2 - {\rm{x}}} \right){\rm{dx}} = {\rm{\lambda }}\mathop \smallint \limits_1^2 \left( { - {{\rm{x}}^2} + 3{\rm{x}} - 2} \right){\rm{dx}}\) 

\(= {\rm{\lambda }} \times \left[ { - \frac{{{{\rm{x}}^3}}}{3} + \frac{{3{{\rm{x}}^2}}}{2} - 2{\rm{x}}} \right]_1^2 = {\rm{\lambda }} \times \left[ { - \frac{{{2^3} - {1^3}}}{3} + \frac{{3\left( {{2^2} - {1^2}} \right)}}{2} - 2\left( {2 - 1} \right)} \right] = {\rm{\lambda }} \times \frac{1}{6}\) 

Now, to be a valid probability density function \(\mathop \smallint \limits_{ - \infty }^\infty {\rm{f}}\left( {\rm{x}} \right){\rm{dx\;\;must\;be\;equal\;to\;}}1.\)

Hence, \({\rm{\lambda }} \times \frac{1}{6} = 1{\rm{\;\;}}\therefore {\rm{\;\lambda }} = 6\)