# Probability of $\lambda$ given k, in Poisson distribution.

We know the PMF of poisson distribution is $P_K(k) = e^{-\lambda} \frac{\lambda ^k}{k!}$, now, given $k$ arrivals in a unit time, what is the PDF of the arriving rate being $\lambda$?

-

I don't know this notation: What does $K$ stands for? – math Dec 4 '12 at 13:48
It means $P(K=k)$, $K$ stands for the random variable, $k$ stands for the value of $K$. – CravingSpirit Dec 4 '12 at 14:07

## 1 Answer

There's no such thing unless you specify a prior. Wikipedia has a section that gives a conjugate prior for this problem.

-

Thanks. This is kind of deep for me at the moment. – CravingSpirit Dec 4 '12 at 14:08
@CravingSpirit: OK; to put it in more mundane terms: What you think about $\lambda$ after you learn that there were $k$ arrivals in a unit time depends on what you that about $\lambda$ before you learned that. For instance, if you think a priori that very low values of $\lambda$ are very unlikely, then even after you observe $0$ arrivals the probability you assign a posteriori to low values of $\lambda$ will be less than the probability you would have assigned a posteriori after observing $0$ arrivals if you had a priori believed these low values to be quite likely. – joriki Dec 4 '12 at 14:30
Ok, you mean the probability of $\lambda$ depends both on our a priori judgment and the value of $k$. – CravingSpirit Dec 4 '12 at 14:35
@CravingSpirit: Yes, that just about summarizes it :-) – joriki Dec 4 '12 at 14:35
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 18, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9161763191223145, "perplexity_flag": "middle"}