Let $X(t)$ be a pure death process starting from $X(0)=N$. Assume that the death parameters are $\mu_1, \mu_2,\dots,\mu_N$. Let $T$ be an independent exponentially distributed random variable with parameter $\theta$. Show that $Pr\{X(T)=0\} = \prod_{i=1}^{N} \frac{\mu_i}{\mu_i+\theta}$.
My thoughts are that I should condition on $T=t$ and integrate so we would have
$Pr\{X(T)=0\} = \int_0^\infty Pr\{X(t) = 0 | T = t\} Pr\{T=t\}dt$
In the book we are given the formula for $P_n(t) = Pr\{X(t) = n\}$ but it is messy, here it is, $P_n(t) = \mu_{n+1}\dots\mu_{N} [ A_{n,n} e^{-\mu_n t} + \dots + A_{N,n}e^{-\mu_N t} ]$ Where, $A_{k,n} = \frac{1}{(\mu_N - \mu_k) \dots (\mu_{k+1} - \mu_k)(\mu_{k-1} - \mu_k) \dots (\mu_n - \mu_k)}$ Using my approach leads to some very ugly algebra and some terms that just don't seem to cancel. I tried doing it for the case of $N=2$ but still could not get it in the right form. I feel like I might be approaching this problem wrong but I don't see any other way to go about it.