I need to calculate the characteristic function of an exponential distribution:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\phi _X \left( t \right) = \int\limits_{ - \infty }^\infty {e^{itX} \lambda e^{ - \lambda x} dx} = \int\limits_{ - \infty }^\infty {\lambda e^{\left( {it - \lambda } \right)x} dx}

[/tex]

I have arrived at the following expression:

[tex]

\frac{{i\lambda }}{{i\lambda + t}}\mathop {\lim }\limits_{x \to \infty } \left( {e^{\left( {\lambda - it} \right)x} } \right)

[/tex]

and I can't calculate the limit.

Any help would be appreciated.

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# Characteristic function of an exponential distribution

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