# Coxian distribution: probability first phase smaller than x

I'm trying to solve a question about the coxian distribution. There are two phases, phase 1 and phase 2. Because coxian is not so famous, I will give the definition as in our book:

Let $$B(x)~\sim~C_2(\alpha,\mu_1,\mu_2)$$. This means that the random variable $$X$$ can be represented as $$\begin{array}{ll} X = X_1+X_2 & \mbox{with probability \alpha}\\ X = X_1 & \mbox{with probability 1-\alpha }\end{array}$$ where $$X_1$$ and $$X_2$$ are independent random variables having exponential distributions with means $$1/\mu_1$$ and $$1/\mu_2$$

Now I want to solve the question: what is the probability that the first phase is below T, so $$P(X_1

My first thought was that $$P(X_1, because $$X_1$$ is exponentially distributed. But maybe this is too simple. Hope one of you can help me...

• The definition of $X$ is clear, what tt's not clear is what's "the first phase" – leonbloy Apr 9 at 16:56
• Hello Leonbloy, the first phase is $X_1$, the first exponential random variable. :) – MathMe Apr 10 at 8:32