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

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 $k\in\{0,1,2,..\}$
CDFSee below

The CDF of the Hypergeometric distribution is given by the following equation:- \begin{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 \end{equation}
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$\Large 1-exp\{-(\frac{x}{\lambda})^k\}$

28 comments:

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  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..
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    ReplyDelete

  11. 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:
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    ReplyDelete
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  13. 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.

    ReplyDelete

  14. 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:
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    ReplyDelete
  15. 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

    ReplyDelete
  16. 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 

    ReplyDelete

  17. 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.

    ReplyDelete
  18. Hello Everyone out there,I am here to give my testimony about a Herbalist doctor who helped me . I was infected with HERPES SIMPLEX VIRUS in 2011, i went to many hospitals for cure but there was no solution, so I was thinking how can I get a solution out so that my body can be okay. One day I was in the river side thinking where I can go to get solution. so a lady walked to me telling me why am I so sad and i open up all to her telling her my problem, she told me that she can help me out, she introduce me to a doctor who uses herbal medication to cure HERPES SIMPLEX VIRUS and gave me his email, so i mail him. He told me all the things I need to do and also give me instructions to take, which I followed properly. Before I knew what is happening after two weeks the HERPES SIMPLEX VIRUS that was in my body got vanished . so if you are also heart broken and also need a help, you can also email him at {oliha.miraclemedicine@gmail.com}
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    5. Hepatitis B
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    7. Emphysema
    8. COPD (Chronic Obstructive Pulmonary Disease

    ReplyDelete
  19. 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 or vist his website https://oyaguspellcaster.wixsite.com/oyaguherbalhome 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.

    ReplyDelete
  20. 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 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 his 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.. or vist his website https://oyaguspellcaster.wixsite.com/oyaguherbalhome
    He can still be able to help you with this herbs medicine:
    1...ALS CURE/DIABETES CURE/EPILESY/HPV CURE/LUPUS/HEPATITIS/CANCER/GOUT

    ReplyDelete
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  24. 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 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 his 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.. or vist his website https://oyaguspellcaster.wixsite.com/oyaguherbalhome. He can still be able to help you with this herbs medicine: 1...ALS CURE/DIABETES CURE/EPILESY/HPV CURE/LUPUS/HEPATITIS/CANCER/GOUT

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  25. I has suffered for Human papillomavirus HPV) for 2years, I was given some tablets at the hospital but I refused to take it, They said I have to be on it for life so I don't want take a drugs everyday for life. No point in taking medicine everyday when u won't get cure from it and I was advice to seek for natural herbal cure, after some time I found dr onokun is the most trustful herbalist that have herbs to cure wicked symptom's,I emailed dr onokun, for 2weeks been his patient he cured my (HPV) with his herbal. I only used his natural herbs for two weeks it was 100% cure. I'm not (HPV) patient anymore. I'm happy about it i finally got cured out of this mess been in my body for 2years. I also recommend you if you're living with (HPV) or herpes symptoms i also want you to be free contact dr onokun with the email attach to my post. Dronokunherbalcure@gmail.com or website www.Dronokunherbalcure.wordpress.com or facebook: https://www.facebook.com/naturaltreatmentcenter1/

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  26. DR OYAGU herbal medicine is a good remedy for herpes simplex virus cure. My name is MIGNON WRIGHT , am from USA,I am here to give my testimony about DR OYAGU who helped me in my life, i want to inform the public how i was cured from (HERPES SIMPLEX VIRUS) by DR OYAGU, i visited different hospital but they gave me list of drugs like Familiar, Zovirax, and Valtrex which is very expensive to treat the symptoms and never cured me. I was browsing through the Internet searching for remedy on HERPES and i saw comment of people talking about how DR OYAGU cured them, when i contacted him he gave me hope and send a Herbal medicine to me that i took and it seriously worked for me, am a free person now without any problem, my HERPES result came out negative, I pray to God to never let me and my family forget you in our life, am now cured permanently, you can also get your self cured my friends if you really need this doctor help, you can reach him on his website ;https://oyaguspellcaster.wixsite.com/oyaguherbalhome Email: oyaguherbalhome@gmail.com or via whatsapp +2348101755322

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  27. 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 or visit his website https://oyaguspellcaster.wixsite.com/oyaguherbalhome 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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  28. Hello Everyone out there,I am here to give my testimony about a Herbalist doctor who helped me . I was infected with HERPES SIMPLEX VIRUS in 2011, i went to many hospitals for cure but there was no solution, so I was thinking how can I get a solution out so that my body can be okay. One day I was in the river side thinking where I can go to get solution. so a lady walked to me telling me why am I so sad and i open up all to her telling her my problem, she told me that she can help me out, she introduce me to a doctor who uses herbal medication to cure HERPES SIMPLEX VIRUS and gave me his email, so i mail him. He told me all the things I need to do and also give me instructions to take, which I followed properly. Before I knew what is happening after two weeks the HERPES SIMPLEX VIRUS that was in my body got vanished . so if you are also heart broken and also need a help, you can also email him at {oliha.miraclemedicine@gmail.com}
    or whatsapp him number: +2349038382931. or website is https://olihamiraclemedicine.webs.com/
    Contact him today and you will have a testimony...Good luck!

    Dr. OLIHA Also Cures:
    1. HIV/AIDS
    2. HERPES 1/2
    3. CANCER
    4. ALS (Lou Gehrig’s disease)
    5. Hepatitis B
    6. chronic pancreatic
    7. Emphysema
    8. COPD (Chronic Obstructive Pulmonary Disease

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