Question

Consider a random sample (X₁, X₂, ..., Xₙ) of size n from a geometric distribution with parameter p. Show that the random variable Y = X₁ + X₂ + ⋯ + Xₙ is a sufficient statistic for p.

Answer 1

The random variable Y = X₁ + X₂ + ... + Xₙ is a sufficient statistic for the parameter p in the geometric distribution.

Step-by-step explanation:

We can show that the random variable Y = X₁ + X₂ + ... + Xₙ is a sufficient statistic for the parameter p by using the factorization theorem. According to the factorization theorem, Y is a sufficient statistic if and only if the joint probability mass function (pmf) of the sample (x₁, x₂, ..., xₙ) can be expressed as the product of two functions:

1. A function that depends only on the sample (x₁, x₂, ..., xₙ), which we'll call g(y; x₁, x₂, ..., xₙ)
2. A function that depends only on the parameter p, which we'll call h(y; p)

Since we are dealing with a geometric distribution, the probability mass function for a single observation, Xᵢ, is given by:

f(x; p) = p(1-p)^(x-1)

By using the properties of the probability mass function, we can rewrite the joint probability mass function of the sample as:

f(x₁, x₂, ..., xₙ; p) = p^n * (1 - p)^(sum(x) - n)

This expression can be factored into two functions: g(y; x₁, x₂, ..., xₙ) = 1 and h(y; p) = p^n * (1 - p)^(sum(x) - n).

Therefore, we can conclude that the random variable Y = X₁ + X₂ + ... + Xₙ is a sufficient statistic for the parameter p in the geometric distribution.