Do you have descriptive/typical examples for processes whose results are described by a beta distribution? So far i only have one:
You have a population of constant size with N individuals and you observe a single gene (or gene locus).
The descendants in the next generation are drawn from a binomial distribution, so some individuals have several descendants, others have no dexcendants.
The gene can mutate at a rate u (for example blue eyes become brown eyes in 10^-5 cases you draw an individual with blue eyes).
Thae rate at which brown eyed individuals have blue eyed descendants ist the same.
The beta distribution describes how likely it is to find X% of the individuals having a ceratian eye colour. Thereby 2*N*u is the value for both parameters of the beta distribution.
Do you have mor examples. For which things is the beta distribution used?
Sven
Best Answer
Completely elementary is the fact that for every positive integers $k\le n$, the distribution of the order statistics of rank $k$ in an i.i.d. sample of size $n$ uniform on the interval $(0,1)$ is beta $(k,n-k+1)$.
Slightly more sophisticated is the fact that, in Bayesian statistics, beta distributions provide a simple example of conjugate priors for binomial proportions. If $X$ conditionally on $U=u$ is binomial $(n,u)$ for every $u$ in $(0,1)$, then the distribution of $U$ conditionally on $X=x$ which is called the conjugate prior of the binomial is beta $(x,n-x)$. This result is a special case of the multinomial Dirichlet conjugacy.
Still more sophisticated is the fact that beta distributions are stationary distributions of Dubins-Freedman processes. These are Markov chains $(X_t)$ on $(0,1)$ moving from $X_t=x$ to $X_{t+1}=xU_t$ with probability $p$ and to $X_{t+1}=x+(1-x)U_t$ with probability $1-p$, where $p$ is a fixed parameter in $(0,1)$ and the sequence $(U_t)$ is an i.i.d. sequence with values in $(0,1)$. If the distribution of $U_t$ is uniform on $(0,1)$, then $(X_t)$ is ergodic and its stationary distribution is beta $(1-p,p)$. The seminal paper on the subject is due to Dubins and Freedman in the Fifth Berkeley Symposium. Later on, Diaconis and Freedman wrote a very nice survey. And the specific result mentioned above was somewhat generalized here.