Physics Forums

## Cannot tell when a probability generating function converges for |s|<1

Hi, I have a problem that is already solved... I thought 3 of the 4 functions were probability generating functions, but I got one wrong and don't know why.

The solution says $g(s)=1+s-s^2$ is not a probability generating function. However, g(1)=1 and I think g(s) converges to 1 for |s|<1. Isn't that correct? If so, what is it that invalidates this function as a probability generating function?

The solution says that $g(s)=(1/3)*(1+s+s^4)$ and $g(s)=(2-s^2)^{-1}$ are prob. gen. functions and that $g(s)=1+s-s^2$ and $g(s)=(1/2)(1+s+s^3)$ are not (I know the last one is not because g(1)=3/2).

Thanks a lot for your help! :) And if you could also give me an explanation of other things I need to look for in a function to tell if it is a p.g.f., I would really appreciate that.

PhysOrg.com  science news on PhysOrg.com  >> Front-row seats to climate change>> Attacking MRSA with metals from antibacterial clays>> New formula invented for microscope viewing, substitutes for federally controlled drug

If $1 + s - s^2$ were a probability generating function, then we would have $P(2) = -1$.

Thread Tools

|                                                                                             |                                            |         |
|---------------------------------------------------------------------------------------------|--------------------------------------------|---------|
| Similar Threads for: Cannot tell when a probability generating function converges for |s|<1 |                                            |         |
| Thread                                                                                      | Forum                                      | Replies |
|                                                                                             | Precalculus Mathematics Homework           | 3       |
|                                                                                             | Set Theory, Logic, Probability, Statistics | 6       |
|                                                                                             | Calculus & Beyond Homework                 | 1       |
|                                                                                             | Calculus & Beyond Homework                 | 7       |
|                                                                                             | Calculus & Beyond Homework                 | 3       |
{"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": 7, "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": 0.9229505658149719, "perplexity_flag": "middle"}