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From literature we know: If a number $n \le x$ is chosen at random, and choose $\lambda \ge 0$ and $j$ not too large (say $\lambda ,j \le 20$) then the number of primes in $[ n , n + \log(n) ]$ is approximately Poisson-distributed (ref here).

Question: what is then (approx.) the distibution of $\log p$ of a type?

$p$ denotes a prime.

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What is $p$, you didn't define it. Are you asking what is the distribution for log of a Poisson random variable? – gt6989b Jun 27 '13 at 13:16
$p$ is prime. To your question, yes, can you help? – al-Hwarizmi Jun 27 '13 at 13:20
up vote 2 down vote accepted

Use $\mathcal{P}$ to denote the Poisson distribution and $\mathbb{P}[A]$ to denote the probability of some event $A$.

Let $X \sim \mathcal{P}(\lambda)$ with cdf $F_X(x) = \mathbb{P}[X \leq x]$ and let $L = \ln(X)$. Then, $$ F_L(x) = \mathbb{P}[L \leq x] = \mathbb{P}[\ln(X) \leq x] = \mathbb{P}[X \leq e^x] = F_X(e^x). $$

Since we know $X \sim \mathcal{P}(\lambda)$, we can plug in: $$ F_L(x) = F_X(e^x) = e^{-\lambda} \sum_{k=0}^{\lfloor e^x \rfloor} \frac{\lambda^k}{k!}, $$ which is enough to completely determine the distribution.

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thanks. To avoid any doubts, would you define what are $\mathcal{P}$ and $\Bbb P$? – al-Hwarizmi Jun 27 '13 at 15:13
@al-Hwarizmi added first sentence notation clarification. – gt6989b Jun 27 '13 at 15:28
thanks. really great help! – al-Hwarizmi Jun 27 '13 at 15:34

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