Question

State the characteristic function for the continuous random variable.

Answer 1

The characteristics function for a random variable X with probability mass function ( p.m.f)/ probability density function (p.d.f) fₓ is defined as t ∈ ℝ as
$\Phi_x(t)=\mathbb{E}_x[e^{itx}]\\=\mathbb{E}_x[cos(tX)+isin(tX)]$
When X is continuous random variable ,
$\Phi_x(t) = \int\limits^{\infty}_{-\infty}{e^{itX}f_X(x)}\,dx$
= $\Phi_x(t) = \int\limits^{\infty}_{-\infty}{cos(tX)f_X(x)}\,dx+i\int\limits^{\infty}_{-\infty}{sin(tX)f_x(x)}\,dx$