Let $p_{x}(x)$ be a probability mass function of a random variable X defined on the positive integers which satisfies the recursive relation:
$$p_{x}(x+1) = \frac{\theta}{x + 1}p_{x}(x)$$
Where $\theta > 0$ and $p_{x}(1)$ is fixed
Determine explicitly $p_{x}$ as a function of $x$ and $\theta$ only. The function found should not depend on $p_{x}(1)$.
I am genuinely at my wits end here. I calculated $p_{x}$ for $x = 1, 2, 3, 4, 5$ and tried playing around with algebra and substitution to no avail. I have like 3 sheets of paper full of fruitless attempts. I've never done anything like this before and could really use a nudge in the right direction. Thank you for your time.
HINT
Write $p_x(1) = \lambda$ for now. We don't know what $\lambda$ is but we will solve for it later.
[Step 1] Now write down $p_x(2), p_x(3),$ etc using the recurrence. Can you find a formula for $p_x(n)$ in terms of only $n, \theta$ and $\lambda$?
[Step 2] Assuming you can do Step 1, there seems to be two parameters, $\theta$ and $\lambda$. $\theta$ is a given, so you need to find out what $\lambda$ is. Now this is a probability distribution, and every $p_x(n)=$ the probability $Prob(X=n)$. There is a very basic constraint on probabilities $p_x(n): n=1, 2, \dots$ that help you solve for $\lambda$.
Can you finish from here or do you need another hint?