Final answer:
To find the maximum likelihood estimator for θ in the given probability model, we need to maximize the likelihood function by taking the derivative with respect to θ and setting it equal to zero. Solving this equation will give us the maximum likelihood estimator for θ.
Step-by-step explanation:
The probability model p(x;θ)=θ²ˣ e^(-θ²/x!) represents a random sample drawn from a probability distribution. To find the maximum likelihood estimator for θ, we need to maximize the likelihood function. The likelihood function is the product of the probabilities of the observed sample values given the parameter θ. In this case, we have a sample of size n, so the likelihood function will be the product of θ^x times e^(-θ²) for each value of x in the sample.
To maximize this function, we take the derivative with respect to θ and set it equal to zero. Solving this equation will give us the maximum likelihood estimator for θ.
For example, if we have a sample of size n=3 with observed values of x={1, 2, 3}, the likelihood function will be θ^1 * θ^2 * θ^3 * e^(-θ²). Taking the derivative and solving for θ will give us the maximum likelihood estimator for θ.