## Wednesday 27 November 2013

### Probability Distributions Tables

Below are some tables detailing the Density Equation, parameters, support and Cumulative Distribution Function (CDF) for a few commonly used probability distributions.

In the tables that follow, the error function $erf(x)$ is $$erf(x) = \frac{2}{\sqrt{\pi}}\int_0^xexp(-t^2)dt \nonumber$$ the Gamma function $\Gamma(a)$ is the following integral $$\Gamma(a)=\int_0^\infty t^{(a-1)}e^{-t}dt \nonumber$$ the lower incomplete Gamma function $\gamma(a,x)$ is $$\gamma(a,x)=\int_0^x t^{(a-1)}e^{-t}dt \nonumber$$ The normalised (or regularized) incomplete Beta function $I_x(a,b)$ is $$I_x(a,b)=\frac{B_x(a,b)}{B(a,b)} \nonumber$$ where $x\leq 1$, and $B(a,b)$ and $B_x(a,b)$ are (respectively) $$B(a,b)=\int_0^1t^{(a-1)}(1-t)^{(b-1)}dt \nonumber$$ and $$B_x(a,b)=\int_0^xt^{(a-1)}(1-t)^{(b-1)}dt \nonumber$$

 Distribution Beta Density Equation $\Large \frac{x^{(\alpha - 1)}(1-x)^{\beta-1}}{B(\alpha,\beta)}$ Parameters $\alpha>0,\beta>0$ Support $0\leq x \leq 1$ CDF $I_x(\alpha,\beta)$

 Distribution Binomial Density Equation ${n \choose k} p^k(1-p)^{n-k}$ Parameters $n \in \{0,1,2,..\},0< p < 1$ Support $k \in\{0,1,2,..\}$ CDF $I_{1-p}(n-k,1+k)$

 Distribution Cauchy-Lorenz Density Equation $\Large \frac{1}{\pi\gamma[1+{(\frac{x-x_0}{\gamma})}^2]}$ Parameters $-\infty < x_0 < +\infty,\gamma > 0$ Support $-\infty < x < +\infty$ CDF $\frac{1}{\pi}atan(\frac{x-x_0}{\gamma})+\frac{1}{2}$

 Distribution Chi-squared Density Equation $\Large \frac{1}{2^{k/2}\Gamma(k/2)}x^{\frac{k}{2}-1}e^{-x/2}$ Parameters $k \in \{0,1,2,..\}$ Support $0\leq x < \infty$ CDF $\Large \frac{1}{\Gamma(k/2)}\gamma(\frac{k}{2},\frac{x}{2})$

 Distribution F-distribution Density Equation $\Large \frac{\sqrt{\frac{(d_1x)^{d_1}d_2^{d_2}}{(d_1x+d_2)^{d_1+d_2}}}}{xB(d_1/2,d_2/2)}$ Parameters $d_1>0,d_2>0$ Support $0\leq x<\infty$ CDF $\Large I_{\frac{d_1x}{d_1x+d_2}}(\frac{d_1}{2},\frac{d_2}{2})$

 Distribution Gamma Density Equation $\Large \frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}$ Parameters $\alpha>0,\beta>0$ Support $0\leq x<\infty$ CDF $\Large \frac{1}{\Gamma(\alpha)}\gamma(\alpha,\beta x)$

 Distribution Gaussian Density Equation $\Large \frac{1}{\sigma\sqrt{2\pi}}exp(-\frac{(x-\mu)^2}{2\sigma^2})$ Parameters $-\infty<\mu<\infty,\sigma>0$ Support $-\infty < x < \infty$ CDF $\Large \frac{1}{2}(1+erf(\frac{x-\mu}{\sigma\sqrt{2}}))$

 Distribution Hypergeometric Density Equation $\Large \frac{{r \choose k}{N-r \choose n-k}}{{N \choose n}}$ Parameters $0\leq k \leq r$, $0\leq (n-k) \leq (N-r)$ Support $k\in\{0,1,2,..\}$ CDF See below

The CDF of the Hypergeometric distribution is given by the following equation:- $$CDF = 1 - \frac{{n \choose k+1}{N-n \choose r-k-1}}{{N \choose r}}(1+\sum_{m=1}^{\infty}\frac{(k+m-r)(k+m-n)}{(k+m+1)(N+k+1+m-r-n)}\frac{1}{(m - 1)!}) \nonumber$$
 Distribution Laplace Density Equation $\Large \frac{1}{2b}exp(-\frac{|x-\mu|}{b})$ Parameters $-\infty < \mu < \infty,b > 0$ Support $-\infty < x < \infty$ CDF $\Large \frac{1}{2}exp(\frac{x-\mu}{b})$ if $x<\mu$ $\Large 1-\frac{1}{2}exp(\frac{\mu-x}{b})$ if $x\geq \mu$

 Distribution Logistic Density Equation $\Large \frac{e^{-\frac{(x-\mu)}{s}}}{s(1+e^{-\frac{x-\mu}{s}})^2}$ Parameters $-\infty < \mu < \infty,s>0$ Support $-\infty < x < \infty$ CDF $\Large \frac{1}{(1+e^{\frac{x-\mu}{s}})^2}$

 Distribution Log-Normal Density Equation $\Large \frac{1}{x\sqrt{2\pi\sigma^2}}exp(-\frac{(ln(x)-\mu)^2}{2\sigma^2})$ Parameters $\sigma > 0,-\infty < \mu < \infty$ Support $0 < x < \infty$ CDF $\Large \frac{1}{2}(1+erf(\frac{ln(x)-\mu}{\sigma\sqrt{2}}))$

 Distribution Pascal Density Equation ${k+r-1 \choose k}(1-p)^rp^k$ Parameters $0 < p < 1,r > 0$ Support $k\in\{0,1,2,..\}$ CDF $1-I_p(k+1,r)$

 Distribution Poisson Density Equation $\Large \frac{\lambda^k}{k!}e^{-\lambda}$ Parameters $\lambda > 0$ Support $k\in\{0,1,2,...\}$ CDF $\Large \frac{\Gamma(k+1,\lambda)}{k!}$

 Distribution Rayleigh Density Equation $\Large \frac{x}{\sigma^2}exp(-\frac{x^2}{2\sigma^2})$ Parameters $\sigma > 0$ Support $0\leq x < \infty$ CDF $\Large 1-exp(-x^2/2\sigma^2)$

 Distribution Student-t Density Equation $\Large \frac{\Gamma(\frac{v+1}{2})}{\sqrt{v\pi}\Gamma(\frac{v}{2})}(1+\frac{x^2}{v})^{\frac{v+1}{2}}$ Parameters $v\in{1,2,3,..}$ Support $-\infty < x < \infty$ CDF $\Large 1-\frac{1}{2}I_{x(t)}(\frac{v}{2},\frac{1}{2})$ for $t>0$ where $\Large x(t)=\frac{v}{t^2+v}$

 Distribution Weibull Density Equation $\Large \frac{k}{\lambda}(\frac{x}{\lambda})^{(k-1)}exp(-(\frac{x}{\lambda})^k)$ Parameters $\lambda > 0,k > 0$ Support $0\leq x < \infty$ CDF $\Large 1-exp\{-(\frac{x}{\lambda})^k\}$

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9. i already gave up on ever getting cured of HSV2 because i have try many treatment none of them work out for me i have go to different hospital they always tell me same thing there is no cure for herpes when i came across a post about Dr oyagu, in the net from a lady called Angela i contacted him and he reassured me with him herbal medicine which i took according to the way he instructed, that how i was cured. I doubted at first because i have been to a whole lot of reputable doctors, tried a lot of medicines but none was able to cure me. so i decided to listen to him and he commenced treatment, and under two weeks i was totally free from Herpes. i want to say a very big thank you to DR oyagu for what he has done in my life. feel free to leave him a message on email oyaguherbalhome@gmail.com and also WhatsApp him +2348101755322..
He can still be able to help you with this herbs medicine:
1...ALS CURE/DIABETES CURE/EPILESY/HPV CURE/LUPUS/HEPATITIS/CANCER/GOUT

10. i already gave up on ever getting cured of HSV2 because i have try many treatment none of them work out for me i have go to different hospital they always tell me same thing there is no cure for herpes when i came across a post about Dr oyagu, in the net from a lady called Angela i contacted him and he reassured me with him herbal medicine which i took according to the way he instructed, that how i was cured. I doubted at first because i have been to a whole lot of reputable doctors, tried a lot of medicines but none was able to cure me. so i decided to listen to him and he commenced treatment, and under two weeks i was totally free from Herpes. i want to say a very big thank you to DR oyagu for what he has done in my life. feel free to leave him a message on email oyaguherbalhome@gmail.com and also WhatsApp him +2348101755322..
He can still be able to help you with this herbs medicine:
1...ALS CURE/DIABETES CURE/EPILESY/HPV CURE/LUPUS/HEPATITIS/CANCER/GOUT

11. I am so happy, i never believe i will be this happy again in life, I was working as an air-hoster ( cabby crew ) for 3years but early this year, i loose my job because of this deadly disease called Herpes virus (HSV), I never felt sick or have any symptom, till all workers were ask to bring their doctors report, that was how i got tested and i found out that am HSV positive that make me loose my job, because it was consider as an STD and is incurable disease, i was so depress was thinking of committing suicide, till i explain to a friend of mine, who always said to me a problem share is a problem solved, that was how she directed me to Dr Isibor, that was how i contacted him and get the medication from this doctor and i got cured for real, I just went back to my work and they also carry out the test to be real sure and i was negative. Please contact this doctor if you are herpes positive  diseases his email is: drisiborspellhome@gmail.com
or his WhatApp contact +2348110493039

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13. i already gave up on ever getting cured of HSV2 because i have try many treatment none of them work out for me i have be going to different hospital they always tell me same thing there is no cure for herpes when i came across a post about Dr oyagu, in the net from a lady called Rebecca i contacted him and he reassured me with him herbal medicine which i took according to the way he instructed, that how i was cured. I doubted at first because i have been to a whole lot of reputable doctors, tried a lot of medicines but none was able to cure me. so i decided to listen to him and he commenced treatment, and under two weeks i was totally free from Herpes. i want to say a very big thank you to DR oyagu for what he has done in my life. feel free to leave him a message on email oyaguherbalhome@gmail.com and also WhatsApp him +2348101755322..
He can still be able to help you with this herbs medicine:
1...ALS CURE/DIABETES CURE/EPILESY/HPV CURE/LUPUS/HEPATITIS/CANCER/GOUT

14. Greatest thanks to Dr Oyagu for his herbal drugs that he prepared for me and when i start using it in just 2weeks i was completely cured and that ended my HERPES SIMPLEX 1&2 DISEASE i am so happy and grateful to Dr Oyagu . after reading about him on a testimony of Jason Cash on a blogger. i knew suddenly Dr Oyagu was the right Doctor to cure my HERPES SIMPLEX 1&2 DISEASE. i discuss with Dr Oyagu and he prepared a herbal medicine for me and when it got sent to me in south korean . i used the herbal medicine and 2weeks and i went to check up again. after 15years of suffering from HERPES SIMPLEX 1&2 at last i am smiling once again. Dr Oyagu also has remedy to others disease like COLD SORES,HIV/AIDS,DIABETES.CANCER,HIGH BLOOD PRESSURE AND MANY MORE. I oblige everyone to contact this powerful herbalist Dr Oyagu and be free from your suffering. contact his WhatsApp line: +2348101755322 or his Email:Oyaguherbalhome@gmail.com

15. I found out, that taking herbal treatment is the best to get rid of hpv as soon as i feel the symptoms of hpv is appearing, i took the healing process by contacting Dr onokun for natural treatment it works wonders, amazingly ever since I had the herbal treatment i have not feel these horrible disease anymore and my doc told me the virus is gone, i am glad i finally got cured out from this horrible disease. every hpv or herpes patients should also get in touch with this herbalist Dr to get ride of these disease forever his email address; dronokunherbalcure@gmail.com

16. Hello Am From The UK , I want to write a little testimony about the good work of doctor Oyagu who cured me from Hsv 1 and 2 for just 2 weeks with his herbal medicine, I never believe I can be normal again and have a good life like others I always regretted the day I got diagnose with the virus, I was lost of hope when my doctor told me there is no cure for it but I keep thinking and thinking about my future, if I can have kids of my own well I am so grateful for my helper who get me cured with his herbal medicine, I go online in search of anything that can help me because I can’t deal with it forever so I found this doc Oyagu email on a blog of someone who was cured by him I quickly contact him for help and explain all my pain to him, he told me not to worry about it there is cure for real, I never believe until he send me the herbal medicine when I order for it and I have it within 4 days that is how I took the medicine for 2 week and the following week I go for test just to confirm I was 100% cured from this sickness what a miracle in my life I am so happy right now, you can also get in contact with him if you have such sickness through email address oyaguherbalhome@gmail.com or text him on what's app, phone number +2348101755322 Dr Oyagu also has remedy to others disease like COLD SORES,HIV/AIDS,DIABETES.CANCER,HIGH BLOOD PRESSURE AND MANY MORE. I oblige everyone to contact this powerful herbalist Dr Oyagu and be free from suffering.

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20. HOW I GOT CURED FROM HERPES VIRUSI was diagnosed of HERPES virus and i have tried all I can to get cured but all to no avail,, until I saw a post in a health forum about a herbalist man who prepare herbal medications to cure all kind of diseases including HERPES virus,at first I doubted if it was real but decided to give it a try,when I contacted this herbalist via his email and he prepared a HERPES herbal cure and sent it to me via UPS courier courier service, when I received this herbal medicine, he gave me step by direction on how to apply it,after applying the way I was instructed, I was totally cured of this deadly disease called HERPES, all thanks to Dr Akhanene
this great herbal doctor via Emai drakhanenespellhome@gmail.com. or whatsapp or call him +2348168714427 HE CAN ALSO CURE SICKNESS LIKE{1}HIV/AIDS{2}DIABETES{3}EPILEPSY{4} BLOOD CANCER{5} HPV{6} ALS{7} HEPATITIS{8}LOVE SPELL{9} SICKLE AND ANAEMIA

21. Herbal treatment is 100% guaranteed to cure Herpes Virus, i have tried and it worked perfectly, contact {{Robinsonbuckler11}} @gmail. com,....

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### Logistic Regression Calculator and ROC Curve Plotter

This blog post implements a Logistic Regression calculator for a binary output. Consider a binary outcome response variable \(Y\...