Introduction to Probability · Problem 10.3.6

Markov Chains — Grinstead & Snell

Mathematics — Probability · moment-generating-functions-continuous

Problem

Let XX be a continuous random variable whose characteristic function kX(τ)k_X(\tau) is kX(τ)=e−∣τ∣,−∞<τ<+∞.k_X(\tau) = e^{-|\tau|}, \qquad -\infty < \tau < +\infty. Show directly that the density fXf_X of XX is fX(x)=1π(1+x2).f_X(x) = \frac{1}{\pi(1 + x^2)}.

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