$X$ is a continuous random variable with PDF $$f(x) = c\theta^{|x|} \quad \text{ for } -\infty<x<\infty,$$ where $0<\theta<1$ is a parameter and $c$ is a constant.

Derive and expression for $c$ in terms of $\theta$.

Well we have that $$I = \int^{\infty}\_{-\infty} c\theta^{|x|} \,\rm dx = 1$$

Splitting this integral up:

\begin{align}
I &= \int^{0}\_{-\infty} c\theta^{-x} \, \rm dx + \int^{\infty}\_{0} c\theta^{x} \, \rm dx - c \\
&= \left[-c\theta^{-x}\right]^{0}\_{-\infty} + \left[c\theta^{x}\right]^{\infty}\_{0} - c \\
&= [-c - \lim\_{n\to-\infty} \theta^{-n}] + [\lim\_{n\to\infty} \theta^{n} - c] - c
\end{align}

Recall that $0<\theta<1$ hence both limits are zero so

$$I = -3c = 1 \iff c = -\frac{1}{3}$$

But this isn't really in terms of theta unless we include $\theta^{0}$ so is this correct?