Question

For each given \(p\), let \(X\) have a binomial distribution with parameters \(p\) and N. Suppose \(P\) is distributed according to a beta distribution with parameters \(r\) and \(s\). Find the resulting distribution of \(X\). When is this distribution uniform on \(x=9,1, \ldots, N ?\)

Short answer

The resulting distribution of \(X\) is given by: \[P(X=k) = \binom{N}{k} \frac{\Gamma(r+s)}{\Gamma(r)\Gamma(s)} B(a, b)\] where \(a = k + r\) and \(b = N - k + s\), and \(B(a, b)\) is the beta function. The distribution is uniform on \(x = 1, 2, \ldots, N\) when there exists a relationship between \(r\) and \(N\) such that: \[\frac{B(1+r, N-1+r)}{B(N+r, r)} = N\] In general, as the value of \(r\) increases relative to \(N\), the distribution becomes closer to uniform.

Step-by-step solution

Law of Total Probability

We have two random variables, X and P. We're interested in the probability distribution of X given that P has a beta distribution. To do this, we use the law of total probability to find the probability mass function of X: \[P(X=k) = \int_0^1 P(X=k|P=p) f_P(p) dp\] where k ∈ {0, 1, ..., N}.

Binomial Distribution

The random variable X follows a binomial distribution with parameters p and N, so its probability mass function is given by: \[P(X=k|P=p) = \binom{N}{k} p^k (1-p)^{N-k}\] where k ∈ {0, 1, ..., N}.

Beta Distribution

The random variable P follows a beta distribution with parameters r and s, so its probability density function is given by: \[f_P(p) = \frac{\Gamma(r+s)}{\Gamma(r)\Gamma(s)} p^{r-1} (1-p)^{s-1}\] where Γ represents the gamma function, and p ∈ (0, 1).

Resulting Distribution of X

Now, we plug in the expressions for the binomial and beta distributions into the law of total probability: \[P(X=k) = \int_0^1 \binom{N}{k} p^k (1-p)^{N-k} \frac{\Gamma(r+s)}{\Gamma(r)\Gamma(s)} p^{r-1} (1-p)^{s-1} dp\] Simplify the expression: \[P(X=k) = \binom{N}{k} \frac{\Gamma(r+s)}{\Gamma(r)\Gamma(s)} \int_0^1 p^{k+r-1} (1-p)^{N-k+s-1} dp\] With the beta function, B(a, b), we have: \[B(a, b) = \int_0^1 p^{a-1} (1-p)^{b-1} dp = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}\] Now, let a = k + r and b = N - k + s, so the integral becomes the beta function B(a, b): \[P(X=k) = \binom{N}{k} \frac{\Gamma(r+s)}{\Gamma(r)\Gamma(s)} B(a, b)\]

Uniform Distribution Condition

The probability mass function is uniform when P(X=k) = constant for k = 1, 2, ..., N. We need to find the values of r and s that make it true. From the expression of P(X=k) above, note that if r = s, then: \[P(X=k) = \binom{N}{k} \frac{\Gamma(2r)}{\Gamma^2(r)} B(k+r, N-k+r)\] Setting the expression of P(X=k) for k = 1 equal to that for k = N, we get: \[\binom{N}{1} \frac{\Gamma(2r)}{\Gamma^2(r)} B(1+r, N-1+r) = \binom{N}{N} \frac{\Gamma(2r)}{\Gamma^2(r)} B(N+r, r)\] Simplifying and rearranging, we get: \[\frac{B(1+r, N-1+r)}{B(N+r, r)} = N\] This equation gives the relationship between r and N when the resulting distribution of X is uniform. In general, as the value of r increases relative to N, the distribution becomes closer to uniform.

Key concepts

Law of Total Probability

The law of total probability is a fundamental rule that relates marginal probabilities to conditional probabilities. It articulates that the probability of an event can be calculated by summing the probabilities of mutually exclusive and exhaustive cases associated with that event.

For a probability distribution, such as binomial in our exercise, we use the law to combine the probabilities across the different values of another, related random variable—in this case, the beta-distributed variable P.

This technique uses integration to add up all the possible influences P can have on X, weighted by P's distribution, to yield the overall probability mass function for X. This is why the solution involves integrating the probability of X given P, multiplied by the probability density function of P.

Probability Mass Function

The probability mass function (PMF) is a key concept when working with discrete random variables, such as those following a binomial distribution. It assigns a probability to each possible outcome of the random variable. Specifically, for a binomial random variable X with parameters p and N, the PMF is given by the formula \[P(X=k|P=p) = \binom{N}{k} p^k (1-p)^{N-k}\]where k can take on any integer value from 0 to N, inclusive. Here, \(\binom{N}{k}\) is the binomial coefficient, representing the number of ways to choose k successes from N trials.

Understanding the PMF allows us to assess the likelihood of different outcomes for X given a specific probability p, which is the basis for finding the overall probability distribution when p itself is a random variable, as it is when P follows a beta distribution.

Gamma Function

The gamma function, represented by \(\Gamma\), extends the concept of factorials to continuous and complex numbers. For positive integers, \(\Gamma(n) = (n-1)!\), which connects it back to the binomial coefficients found in binomial distribution formulas.

In probability and statistics, the gamma function is used to describe the probability density functions of various continuous distributions, such as the beta distribution. For example, the beta distribution's density function includes the term \[\frac{\Gamma(r+s)}{\Gamma(r)\Gamma(s)}\]in which the gamma function helps to normalize the distribution, ensuring that the total probability across the entire range of P (in this case, 0 to 1) equals 1.

Beta Function

The beta function, denoted B(a, b), is closely related to the gamma function and is key to solving the integral present in our problem. It is defined as: \[B(a, b) = \int_0^1 p^{a-1} (1-p)^{b-1} dp = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}\]This function represents the integral of x raised to the power of a-1 times (1-x) raised to the power of b-1 over the interval from 0 to 1.

In the context of finding the distribution of X, we use the beta function to evaluate the integral in the law of total probability and thus express P(X=k) in a closed form that includes the beta function B(k + r, N - k + s). It is this connection between the gamma and beta functions that enables us to solve for the probability mass function of X given a beta-distributed P.

Uniform Distribution Condition

When we say a distribution is uniform, we mean that all outcomes are equally likely. For a distribution to be uniform, its probability mass function must assign the same probability to each possible outcome.

In the context of our exercise, we're looking for conditions under which the resulting distribution of X is uniform across the values k=0,1,...,N. This occurs when the probability mass function P(X=k) is constant, irrespective of k. Through algebraic manipulation and by setting r equal to s, we are able to derive a condition where the beta function takes on forms that produce equal probabilities. This is an example of how parameters of one distribution can influence the form of another when related through laws such as the law of total probability.