Question

The number N of eggs laid by a female robin has a Poisson distribution with mean μ. Each egg has probability θ of hatching, independently of other eggs. Given that n eggs were laid, the number Y which hatch has a binomial (n,θ) distribution. (a) (2 points) Find the joint probability mass function of N and Y,f N,Y ​ (n,y;μ,θ). (b) (4 points) A biologist records n i ​ , the number of eggs laid, and y i ​ , the number which hatch, for i=1,2,…,K female robins. Find the log-likelihood function and the joint MLE( μ ^ ​ , θ ^ ).

Answer 1

(a) The joint probability mass function of
`\(N\)` and
`\(Y\)` is given by
`\(f_(N,Y)(n, y; \mu, \theta) = (e^(-\mu)\mu^n)/(n!) \binom{n}{y} \theta^y (1-\theta)^(n-y).\)`

(b) The log-likelihood function for
`\(K\)` observations is
`\(\log\mathcal{L}(\mu, \theta) = \sum_(i=1)^(K)\left(y_i\log(\theta) + (n_i - y_i)\log(1-\theta) - (\mu)/(n_i)\right).\)`

The joint maximum likelihood estimates
`(\(\hat{\mu}, \hat{\theta})\)` can be found by maximizing this log-likelihood function.

Step-by-step explanation:

(a) The probability mass function (PMF) of a Poisson distribution is
`\(P(N = n) = (e^(-\mu)\mu^n)/(n!),\)` and the PMF of a binomial distribution is
`\(P(Y = y | n, \theta) = \binom{n}{y} \theta^y (1-\theta)^(n-y).\)` Considering the independence of the events, the joint PMF is the product of the individual PMFs, leading to the expression in the final answer.

(b) The log-likelihood function is derived by taking the logarithm of the joint probability mass function. The goal is to find the values of
`\(\mu\)` and
`\(\theta\)` that maximize this function. Taking the partial derivatives with respect to
`\(\mu\)` and
`\(\theta\)`, setting them equal to zero, and solving the resulting system of equations provides the maximum likelihood estimates.

In the log-likelihood function,
`\(\log(\theta)\)` and
`\(\log(1-\theta)\)` appear due to the binomial distribution. The sum is taken over
`\(K\)` independent observations, each contributing to the overall likelihood. The maximum likelihood estimates
`(\(\hat{\mu}, \hat{\theta})\)` are obtained by solving the system of equations formed by the first-order conditions.