Because we're going in with prior expectations.
We know that in history, most batting averages over a season have hovered between something like. We know that if a player gets a few strikeouts in a row at the start, that might indicate he'll end up a bit worse than average, but we know he probably won't deviate from that range.
Given our batting average problem, which can be represented with a binomial distribution a series of successes and failures , the best way to represent these prior expectations what we in statistics just call a prior is with the Beta distribution- it's saying, before we've seen the player take his first swing, what we roughly expect his batting average to be. The domain of the Beta distribution is 0, 1 , just like a probability, so we already know we're on the right track- but the appropriateness of the Beta for this task goes far beyond that. We expect that the player's season-long batting average will be most likely around.
You asked what the x axis represents in a beta distribution density plot- here it represents his batting average. Thus notice that in this case, not only is the y-axis a probability or more precisely a probability density , but the x-axis is as well batting average is just a probability of a hit, after all! The Beta distribution is representing a probability distribution of probabilities.
But here's why the Beta distribution is so appropriate. Imagine the player gets a single hit. His record for the season is now 1 hit; 1 at bat. We have to then update our probabilities- we want to shift this entire curve over just a bit to reflect our new information. While the math for proving this is a bit involved it's shown here , the result is very simple. The new Beta distribution will be:. Notice that it has barely changed at all- the change is indeed invisible to the naked eye!
That's because one hit doesn't really mean anything. However, the more the player hits over the course of the season, the more the curve will shift to accommodate the new evidence, and furthermore the more it will narrow based on the fact that we have more proof.
Let's say halfway through the season he has been up to bat times, hitting out of those times.
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Notice the curve is now both thinner and shifted to the right higher batting average than it used to be- we have a better sense of what the player's batting average is. One of the most interesting outputs of this formula is the expected value of the resulting Beta distribution, which is basically your new estimate. You might notice that this formula is equivalent to adding a "head start" to the number of hits and non-hits of a player- you're saying "start him off in the season with 81 hits and non hits on his record". Thus, the Beta distribution is best for representing a probabilistic distribution of probabilities - the case where we don't know what a probability is in advance, but we have some reasonable guesses.
A Beta distribution is used to model things that have a limited range, like 0 to 1. Examples are the probability of success in an experiment having only two outcomes, like success and failure. If you do a limited number of experiments, and some are successful, you can represent what that tells you by a beta distribution.
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Another example is order statistics. For example, if you generate several say 4 uniform 0,1 random numbers, and sort them, what is the distribution of the 3rd one? I use them to understand software performance diagnosis by sampling. This result shows that the Beta distributions naturally appear in mathematics, and it has some interesting applications in mathematics.
First, the beta distribution is conjugate prior to the Bernoulli distribution. That means that if you have an unknown probability like the bias of a coin that you are estimating by repeated coin flips, then the likelihood induced on the unknown bias by a sequence of coin flips is beta-distributed. Second, a consequence of the beta distribution being an exponential family is that it is the maximum entropy distribution for a set of sufficient statistics.
The beta distribution is not special for generally modeling things over [0,1] since many distributions can be truncated to that support and are more applicable in many cases. Let's assume a seller on some e-commerce web-site receives ratings of which are good and are bad. But the "true" quality in terms of ratings we don't know. The bar plot is the density of the histogram of the result of the simulation.
So far the preponderance of answers covered the rationale for Beta RVs being generated as the prior for a sample proportions, and one clever answer has related Beta RVs to order statistics. Gamma RVs already have their rationale in modeling arrival times for independent events, so I will not address that since it is not your question.
But a "fraction of time" spent completing one of two tasks performed in sequence naturally lends itself to a Beta distribution. You have a two dimensional parameter space one for successes contribution and one for failures contribution which makes it kind of difficult to think about and understand. I don't know what they call the prior assumption of 81 hits and outs but in English, that's the a priori assumption.grupoavigase.com/includes/496/3942-donde-conocer-gente.php
Work What You Got by Stephanie Perry Moore
Notice how as the season progresses the curve shifts left or right and the modal probability shifts left or right but there is still a curve. I wonder if the Laa of Large Numbers eventually takes hold and drives the batting average back to. To guesstimate the alpha and beta in general one would take the complete number of prior occurrences at bats , the batting average as known, obtain the total hits the alpha , the beta or the grand total minus the failures and voila - you have your formula.
Then, work the additional data in as shown. The beta distribution is very useful when you are working with particle size distribution. By the way, what's up if you produce a size distribution from a microscopic observation and you have a particle distribution in number, and your aim is to work with a volume distribution? It is almost mandatory to get the original distribution in number bounded on the right.
So, the transformation is more consistent because you are sure that in the new volume distribution does not appear any mode, nor median nor medium size out of the interval you are working. Besides, you avoid the Greenland Africa effect. The transformation is very easy if you have regular shapes, i.
You ought to add three units to the alpha parameter of the number beta distribution and get the volume distribution. I think there is NO intuition behind beta distribution! The beta distribution is just a very flexible distribution with FIX range! And for integer a and b it is even easy to deal with. Also many special cases of the beta have their native meaning, like the uniform distribution.
So if the data needs to be modeled like this, or with slightly more flexibility, then the beta is a very good choice. Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site the association bonus does not count.
Would you like to answer one of these unanswered questions instead? Sign up to join this community. The best answers are voted up and rise to the top. Among some works about stress-strength reliability based on records, Baklizi [ 2 , 4 ] studied point and interval estimation of the stress-strength reliability using record data in the one and two parameter exponential distributions.
Baklizi [ 3 ] considered the likelihood and Bayesian estimation of stress-strength reliability using lower record values from the generalized exponential distribution. Also in the literature the estimation of R in the case of Weibull, exponential, Inverted exponential, Generalized Lindley, generalized exponential and many other distributions has been obtained.
Some of the recent work on the stress-strength model can be seen in [ 13 , 16 , 17 , 18 ].
Recently Singh et al. P—P plots for data set X left and data set Y right. The authors would like to thank the anonymous referees for careful reading and many helpful suggestions. AI, KFV and MH contributed to the design and implementation of the research, to the analysis of the result and to the writing of the manuscript. Skip to main content Skip to sections. Advertisement Hide. Download PDF. On the estimation of stress strength reliability parameter of inverted gamma distribution.
Open Access. First Online: 06 March Introduction In statistical literature the gamma distribution has been the subject of considerable interest, study, and applications for many years in different areas such as medicine, engineering, economics and Bayesian analysis. In this section, we have developed the Bayesian estimation procedure for the estimation of parameter R from inverted gamma and inverted exponential distributions assuming independent gamma priors for the unknown model parameters.
Lemma 4. In this section we use two real data set to show that the IGa distribution can be a better model than other ones. Table 1 The MLEs of parameters for example 1. Open image in new window. In second example we consider data sets, from two groups of patients suffering from head and neck cancer disease which are initially proposed by [ 6 ]. The data are corresponded to the survival times of 51 patients in one group were treated using radiotherapy X , whereas the 45 patients belonging to other group were treated using a combined radiotherapy and chemotherapy Y.
Based on the values of these statistics, we conclude that the IGa distribution, sometimes is better than and sometimes is as good as others models. In Fig. These figures illustrate again that the IGa distribution has a good fit for data. Table 2 The MLEs of parameters for example 2.
Work What You Got
Acknowledgements The authors would like to thank the anonymous referees for careful reading and many helpful suggestions. Abid, S. Baklizi, A. Data Anal. Birnbaum, Z. Efron, B. Gelen, A. Gelman, A. Bayesian Anal.