The value of K for which f(x) is a valid probability density function is 1/4.
[tex]\int\limits^4_0 {fx(x)} \, dx =1[/tex]
[tex]\int\limits^2_0 {Kx} \, dx +\int\limits^4_2 {K(4-x)} \, dx =1[/tex]
[tex]K[\frac{2^2}{2} -0]+[K[4(4-2)-(\frac{4^2}{2} -\frac{2^2}{2} )]=1[/tex]
open the equation
[tex]K\frac{4}{2}+K[8 - (\frac{16}{2} -\frac{4}{2} )] = 1\\[/tex]
[tex]2K+K[\frac{4}{2} ]=1[/tex]
2K + 2K = 1
4K = 1
divide through by 4
K = 1/4
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