A Generalized Geometric Distribution from Vertically Dependent Bernoulli Random Variables

# A Generalized Geometric Distribution from Vertically Dependent Bernoulli Random Variables

## Abstract

This paper generalizes the notion of the geometric distribution to allow for dependent Bernoulli trials generated from dependency generators as defined in Traylor and Hathcock’s previous work. The generalized geometric distribution describes a random variable $X$ that counts the number of dependent Bernoulli trials until the first success. The main result of the paper is  $X$ can count dependent Bernoulli trials from any dependency structure and retain the same distribution. That is, if  $X$ counts Bernoulli trials with dependency generated by $\alpha_{1} \in \mathcal{C}_{\delta}$, and $Y$ counts Bernoulli trials with dependency generated by $\alpha_{2} \in \mathscr{C}_{\delta}$, then the distributions of  $X$ and $Y$ are the same, namely the generalized geometric distribution. Other characterizations and properties of the generalized geometric distribution are given, including the MGF, mean, variance, skew, and entropy.

## Introduction

The standard geometric distribution counts one of two phenomena:

1.  The count of i.i.d. Bernoulli trials until the first success
2.  The count of i.i.d. Bernoulli trials that resulted in a failure prior to the first success

The latter case is simply a shifted version of the former. However, this distribution, in both forms, has limitations because it requires a sequence of independent and identically distributed Bernoulli trials. Korzeniowski [2] originally defined what is now known as first-kind (FK) dependent Bernoulli random variables, and gave a generalized binomial distribution that allowed for dependence among the Bernoulli trials. Traylor [4] extended the work of Korzeniowski into FK-dependent categorical random variables and derived a generalized multinomial distribution in a similar fashion. Traylor and Hathcock [5] extended the notion of dependence among categorical random variables to include other kinds of dependency besides FK dependence, such as sequential dependence. Their work created a class of vertical dependency structures generated by a set of functions

$$\mathscr{C}_{\delta} = \{\alpha: \mathbb{N}_{\geq 2} \to \mathbb{N} : \alpha(n) < n \text{ and } \forall n \exists j \in \{1,\ldots,n-1\} : \alpha(n) = j\},$$

where the latter property is known as dependency continuity. In this paper, we derive a generalized geometric distribution from identically distributed but dependent Bernoulli random variables. The main result is that the pdf for the generalized geometric distribution is the same regardless of the dependency structure. That is, for any $\alpha \in \mathscr{C}_{\delta}$ that generates a sequence of identically distributed but dependent Bernoulli trials, the generalized geometric distribution remains unchanged.

## 37 thoughts on “A Generalized Geometric Distribution from Vertically Dependent Bernoulli Random Variables”

1. FIFA 19 Coin Generator for unlimited free Coins & Points.Run the FIFA 19 Coin Generator to generate unlimited free FIFA 19 coins and points for Xbox … No more spending money for Points or other items.https://tinyurl.com/y2rpqyvy

2. Make money $$playing games at home.Did you know the average annual income for a video game tester is actually 39,063? That’s more than the average USA 2018 personal income of 35,000.And what most unlucky folks have to do to make that sort of money is nowhere near as pleasant as getting paid to sit on your favorite couch playing video games in the comfort of your own home – like you can be doing just a few hours from now.So, if you haven’t yet checked out this amazing opportunity, don’t you think you owe it to yourself and your loved ones?Just think of all those extra luxuries you’ll be able to afford with that sort of extra income.So go here, right now: https://tinyurl.com/y4p9hkzk and see what the fuss is all about. 3. Great article post.Really looking forward to read more. Much obliged. 4. Thank you for sharing superb informations. Your site is so cool. I’m impressed by the details that you have on this web site. It reveals how nicely you understand this subject. Bookmarked this web page, will come back for extra articles. You, my pal, ROCK! I found simply the information I already searched everywhere and simply could not come across. What a great web-site. 5. Great goods from you, man. I’ve understand your stuff previous to and you’re just extremely wonderful. I really like what you’ve acquired here, certainly like what you’re stating and the way in which you say it. You make it enjoyable and you still care for to keep it wise. I can’t wait to read much more from you. This is really a great web site. 6. I have been examinating out many of your stories and i must say pretty clever stuff. I will surely bookmark your blog. 7. Fast and Easy Ways to Clear Acne Fast – Best Way to Clear Acne Naturally!Acne is a general skin infection that many of us have once suffered or are still suffering one time or another.It is annoying when you look in the mirror, seeing your breakouts daily is a thing that can cause frustration.So when we talk of the best way to clear acne fast we mean the easiest way to get rid of this skin infection.Acne is an infection that causes so much damage to our entire life for example it causes low self esteem, depression, frustration and many other bad emotional feelings.The fastest ways to gain freedom from this infection is by using natural methods or treatment. Here are some information that can help you clear acne in a short time but remember that we all have different type of skin and reactions to t treatment.Some skins are very sensitive to treatment while others are not.Avoid eating foods that contains too creamy, oily and sugary because boost our hormones and leads to hormonal inflammation or change in our hormonal level.Eating food like pizza, milkshake, potato chips and so on might increase your breakout overnight.Drinking lot of water can save you from acne and also try to eat foods that have fibre in it.It helps your digestions system a lot and makes them work properly.Water can help you get rid of toxins in your body.Blueberries andstrawberries also help to rid acne quick, eat more of fruits and veggies they are essential and have natural anti acne properties.You can also mix oatmeal and honey together in a bowl, and apply it to the acne affected areas on your skin for few minutes.Then wash it off with clean water and a mild soap after twenty to thirty minutes.The mixture will help you clean your skin and get rid of the acne causing bacteria it is one of the best way to clear acne fast within a short period of time.To clear acne naturally use natural method because they do not have any unfavorable side effects. So are you are interested in natural ways to get rid of acne?You can read more info here : http://dalink.club/cpoM 8. Best Way To Make Money In GTA 5 Online.Making money in GTA V is actually the primary goal of the game, however, it is a hard thing to do.A gamer who isn’t familiar yet with Grand Theft Auto 5 cheats will most likely to spend a lot of time just to get the cash they need to enhance and develop their own character.However, you don’t want to spend hours and hours just to have in-game money, isn’t it?This game should supposedly be fun and enjoying.That’s why we are here to help you!If you want to know how to make money in GTA 5 and want also to earn unlimited cash to develop your character, our GTA 5 money generator will come to rescue.We recommend you to choose our GTA online money generator so you don’t have to do deadly tasks just to earn in-game money, especially in the online version where you’re interacting and playing with other gamers.Our money generator is the fastest way to make money in GTA Online, 100 satisfaction guaranteed.You need to understand: the generation of money for your character’s development is never will be a problem for you anymore!Take your GTA money NOW and have fun in game here : GTA V LIVE CASH$$\$ GENERATOR

9. I cannot thank you enough for the post.Really looking forward to read more. Cool.

10. Thanks for sharing superb informations. Your site is very cool. I am impressed by the details that you¡¦ve on this website. It reveals how nicely you perceive this subject. Bookmarked this website page, will come back for more articles. You, my friend, ROCK! I found simply the information I already searched all over the place and simply could not come across. What an ideal web site.

11. Canada>Canada meloxicam 15 mg tabs The giant birthday card, 3 meters wide and 4 meters long, has been put on display and its sponsors said they hope to gather 10,000 signatures for the beloved former leader, who is still in critical condition after two months in hospital.

12. INSTAGRAM FOLLOWERS GENERATOR.Become a regional celeb. Get limitless absolutely free followers now and impress your mates!Consider you free followers for Instagram from in this article : http://free-followers-ig.cf/

13. I concur- people better not ignore inequality any longer. So what now?

14. Unlimited Elixir, Gems and Gold for Free! I have been sick and tired of spending my money on virtual resources, so I started to search for a Clash of Clans tool which could give me Unlimited Elixir, Gems and Gold. Luckily, I found an awesome one which is easy to use. You can have Unlimited Elixir, Gems and Gold for Free for Clash of Clans here: http://ogurl.co/cqoE

15. Hello, i believe that i saw you visited my web site thus i
got here to go back the favor?.I am attempting
to find things to enhance my web site!I guess its good enough to

16. I conceive you have noted some very interesting details , regards for the post.

17. Пополение баланса Авито (Avito) за 50 | Телеграмм @a1garant Мое почтение, дорогие друзья! Готовы предоставить Всем вам услуги по пополнению баланса на действующие активные аккаунты Avito (а также, совершенно новые). Если Вам необходимы определенные балансы – пишите, будем решать. Потратить можно на турбо продажи, любые платные услуги Авито (Avito). Аккаунты не Брут. Живут долго. Процент пополнения в нашу сторону и стоимость готовых аккаунтов: 50 от баланса на аккаунте.Если необходим залив на ваш аккаунт, в этом случае требуются логин и пароль Вашего акка для доступа к форме оплаты, пополнения баланса.Для постоянных заказчиков гибкая система бонусов и скидок!Гарантия: И, конечно же ничто не укрепляет доверие, как – Постоплата!!! Вперед денег не просим… А также, гасим Штрафы ГИБДД за 65 … Оплата:ЯндексДеньги, Webmoney (профессиональные счета)Рады сотрудничеству! Залив Авито________ авито петербургакции авитоавито работа свежиескидка турбо авитоавито пермь

18. My partner and I absolutely love your blog and find many of your post’s to be exactly what I’m looking for. Do you offer guest writers to write content for you? I wouldn’t mind publishing a post or elaborating on many of the subjects you write about here. Again, awesome weblog!

19. Will I be paid weekly or monthly? celexa manufacturer coupon Not all of the women had been picked by love-hunters. A few, such as Ruby, were there on the prowl themselves. A so-called “jade princess” (the term for women who have been successful in the jade industry), Ruby runs a family business in southern China and doesn’t quite have the model-like features of the other women taking part in the weekend: a little softer around the tummy, her eyebrows a tad overplucked. At 28, she is considered to be a sheng nu or “leftover woman”, a derogatory term used for unmarried women who are 27 and older.

20. Whats up very nice web site!! Man .. Excellent .. Amazing .. I’ll bookmark your blog and take the feeds also…I’m glad to search out so many useful information right here within the put up, we’d like work out extra techniques on this regard, thanks for sharing.