Doing this exercise:
Iwana Passe is taking a multiple-choice exam. You may assume that the number of questions is infinite. Simultaneously, but independently, her conscious and subconscious faculties are generating answers for her, each in a Poisson manner. (Her conscious and subconscious are always working on different questions.) Conscious responses are generated at the rate λc responses per minute. Subconscious responses are generated at the rate λs responses per minute. Assume λc != λs. Each conscious response is an independent Bernoulli trial with probability pc of being correct. Similarly, each subconscious response is an independent Bernoulli trial with probability ps of being correct. Iwana responds only once to each question, and you can assume that her time for recording these conscious and subconscious responses is negligible.
(d) Determine the probability density function for random variable X, where X is the time from the start of the exam until Iwana makes her first conscious response which is preceded by at least one subconscious response
I'm trying to solve it with such reasoning:
C - time from the beginning until first conscious response
B - at least 1 subconscious response occurred during [0; C] time interval
$$ f_c(c) = λ_c*e^{-λ_c*c} $$
Using Poisson PMF of number of k arrivals in a time interval t: $$ P(B | C = c) = 1 - P(0\ subsconscious\ responses\ occurred\ during\ time\ interval [0; c]) = 1 - e^{-λ_s*c} $$
Thus: $$ f_x(x) = f_c(x)*P(B | C = x) = λ_c*e^{-λ_c*x} * (1 - e^{-λ_s*x} ) $$
But my PDF of X doesn't match the one given in the answer (tested graphically). Why is this reasoning not correct? Answer is attached as image:
If you seek to obtain $f_{X}$ by conditioning on $C$, the formula you should be using is $$f_{X}(x)=\int_{0}^{\infty}f_{X|C}(x|c)f_{C}(c)\mathrm{d}c$$ Here, $f_{X|C}$ denotes the conditional density of $X$ given the arrival time of the first conscious response. It's important to remember that $f_{X|C}$ is not a probability but rather a probability density.
If you're interested in finding the pdf of $X$ through conditioning, here is one way.
Fix any $x\geq 0$ and take $N$ as the total number of answers (either conscious of subconscious) produced on $[0,x)$. Notice $N\sim \text{Poisson}\left((\lambda_c +\lambda_s)x\right)$ and from the total law, $$\mathbb{P}(X>x)=\sum_{n=0}^{\infty}\mathbb{P}(X>x|N=n)\mathbb{P}(N=n)$$ The event that $X>x$ given $N=n$ will only occur if all that conscious responses that occur in $[0,x)$ take place before any subconscious responses. Therefore, $$\mathbb{P}(X>x|N=n)=\sum_{k=0}^n\left(\frac{\lambda_c}{\lambda_c + \lambda_s}\right)^k\left(\frac{\lambda_s}{\lambda_c + \lambda_s}\right)^{n-k}=\frac{\lambda_s^{n+1}-\lambda_c^{n+1}}{(\lambda_s-\lambda_c)(\lambda_s+\lambda_c)^n}$$ We get that $$\mathbb{P}({X}>x)=\sum_{n=0}^{\infty}\frac{\lambda_s^{n+1}-\lambda_c^{n+1}}{(\lambda_s-\lambda_c)(\lambda_s+\lambda_c)^n}\cdot e^{-(\lambda_s + \lambda_c)x}\frac{[(\lambda_s + \lambda_c)x]^n}{n!}$$ Using routine calculus this sum evaluates to $$\mathbb{P}(X>x)=\frac{e^{-(\lambda_s + \lambda _c)x}(\lambda_s e^{\lambda_s x}-\lambda_c e^{\lambda_c x})}{(\lambda_s-\lambda_c)}$$ Taking a derivative and multiplying by $-1$ yields the density $$f_{X}(x)=\frac{\lambda_s \lambda_c}{\lambda_s - \lambda_c}(e^{-\lambda_c x}-e^{-\lambda_s x})$$ which is exactly the pdf provided in the solution key.
Question: I originally attempted to find a formula for $f_{X|C}(x|t)$ for $0 \leq t \leq x$ by using a recursive relation... $$\mathbb{P}(X>x|C=t)=\mathbb{P}(N_s(c)=0)\times \mathbb{P}(X>x-t)$$ where $N_s(c)\sim \text{Poisson}(\lambda_s c)$ is the number of subconscious responses provided in $[0,c)$. This led me to the functional equation $$f_{X}(x)=\int_0^x \lambda_c e^{-(\lambda_c + \lambda_s)t}f_{X}(x-t)\mathrm{d}t$$ Unfortunately for me, $f_{X}(x)=\frac{\lambda_s \lambda_c}{\lambda_s - \lambda_c}(e^{-\lambda_c x}-e^{-\lambda_s x})$ is not a solution to this functional equation. Can anyone help me understand why $\mathbb{P}(X>x|C=t)=\mathbb{P}(N_s(c)=0)\times \mathbb{P}(X>x-t)$ isn't valid? Maybe I'm overlooking something obvious. Thank you.