It Helps us to Find the CDF of a Random Variable Directly from MGF or Characteristic Function .

if $X$ is any Random Variable with Characteristic Function (CF) given by

$$ \Phi_X(\omega)= \int_{-\infty}^{\infty} f_X(x) e^{j \omega x} dx $$, Then

$$ F_X(x)=0.5-\frac{1}{j\,\pi}\int_0^{\infty} \frac{\Phi_X(\omega)}{\omega} d \omega $$

if $X$ is any Random Variable with Characteristic Function (CF) given by

$$ \Phi_X(\omega)= \int_{-\infty}^{\infty} f_X(x) e^{j \omega x} dx $$, Then

$$ F_X(x)=0.5-\frac{1}{j\,\pi}\int_0^{\infty} \frac{\Phi_X(\omega)}{\omega} d \omega $$

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