Question

The geometric distribution represents the number of tries you need to do to get a success in a series of independent Bernoulli trials. For example, the number of times you need to roll a die to get a 6 follows a geometric distribution. The geometric distribution has one parameter, p, which is the probability of success of a single trial.

Suppose that X₁,...,Xₙ ∼Geom(p). Write down the likelihood L(p|x₁,...,xₙ).

Answer 1

The likelihood for n independent geometric random variables with success probability p is the product of the probabilities of each observation, represented as L(p|x1,...,xn) = Πi=1n p(1-p)^(xi-1).

Step-by-step explanation:

The student is asking to write down the likelihood L(p|x₁,...,xₙ) for n independent and identically distributed geometrical random variables X₁,...,Xₙ, which have a common success probability p. The likelihood function represents the probability of observing a particular set of data given parameter values for the model. In the case of geometric random variables, each observation Xᵢ represents the number of trials until the first success, and the probability of exactly Xᵢ - 1 failures before the first success is p(1-p)³(Xᵢ-1). When we have n observed values x₁,...,xₙ, the likelihood is the product of each individual observation's probability.

The likelihood L(p|x₁,...,xₙ) is given by:

L(p|x₁,...,xₙ) = Πᵢ=1ₙ p(1-p)³(xᵢ-1)

This is because the observations are independent, so the joint probability is the product of the individual probabilities.