Harper George

2022-09-06

I have equation for truncated increasing geometric distribution given as $pr=\frac{(1-\alpha ){\alpha}^{CW}}{1-{\alpha}^{CW}}.{\alpha}^{-r}$ with $r=1,.....,CW$. $\alpha $ is parameter we can set between 0 and 1.

Here probability of picking $r=1$ is very low and $r=CW$ is very high. How can i derive an equation from this equation that should give me values generated between $r=1$ and CW with given distribution.

i should get values like this if $r=1$ to 10. $CW=10\text{}10,10,9,9,6,1,6,10,10,3,4$(following geometric distribution from equation provided)

Here probability of picking $r=1$ is very low and $r=CW$ is very high. How can i derive an equation from this equation that should give me values generated between $r=1$ and CW with given distribution.

i should get values like this if $r=1$ to 10. $CW=10\text{}10,10,9,9,6,1,6,10,10,3,4$(following geometric distribution from equation provided)

Mckenna Friedman

Beginner2022-09-07Added 10 answers

Step 1

The standard way to generate a geometric variable with probability mass function $P(n)=(1-p){p}^{n}$ for $n\in {\mathbb{N}}_{0}$ from a variable u uniformly distributed over [0,1] is $n=\lfloor \frac{\mathrm{log}u}{\mathrm{log}p}\rfloor \phantom{\rule{thickmathspace}{0ex}}.$

Step 2

To restrict to $0\le n<CW$, we need to transform u to $[{u}_{0},1]$ such that $\frac{\mathrm{log}{u}_{0}}{\mathrm{log}p}=CW$. Thus ${u}_{0}={p}^{CW}$, so you can use $u\to 1-u(1-{p}^{CW})$, and the formula for generating n becomes

$n=\lfloor \frac{\mathrm{log}(1-u(1-{p}^{CW}))}{\mathrm{log}p}\rfloor \phantom{\rule{thickmathspace}{0ex}}.$

To map this to your case, take $r=CW-n$ and $p=\alpha $

The standard way to generate a geometric variable with probability mass function $P(n)=(1-p){p}^{n}$ for $n\in {\mathbb{N}}_{0}$ from a variable u uniformly distributed over [0,1] is $n=\lfloor \frac{\mathrm{log}u}{\mathrm{log}p}\rfloor \phantom{\rule{thickmathspace}{0ex}}.$

Step 2

To restrict to $0\le n<CW$, we need to transform u to $[{u}_{0},1]$ such that $\frac{\mathrm{log}{u}_{0}}{\mathrm{log}p}=CW$. Thus ${u}_{0}={p}^{CW}$, so you can use $u\to 1-u(1-{p}^{CW})$, and the formula for generating n becomes

$n=\lfloor \frac{\mathrm{log}(1-u(1-{p}^{CW}))}{\mathrm{log}p}\rfloor \phantom{\rule{thickmathspace}{0ex}}.$

To map this to your case, take $r=CW-n$ and $p=\alpha $

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