Question 1:

These data arise from a polyspermy example with sea urchin eggs (Echinus esculentus), with a sperm density of 6x10⁷ sperm/mlat 15°C. A dilute solution of nitrogen was also present.

An alternative mechanism for the process of fertilisation is size-limited with

p_n(t) = (^m_n)e^-λtn(1 - e^-λt)^m-n, 0 ≤ n ≤ m,

where p_n (t) is the probability that at time t after the start of the experiment an egg has n sperm. In this case there is an upper bound, m, on the number of fertilisations that can take place. Consider what underlying mechanism results in such a model. Write down the likelihood corresponding to the data from an experiment of one duration only, and obtain the maximum likelihood estimator of the single parameter λ in this case. Examine the goodness of fit of the model.

Question 2:

Define X -Bin(n, p) and p-Be(α, β). Derive the probability mass function of the beta-binomial distribution and write it in the form

Pr(X = 0) = Π_j=0^n-1 (1 - μ + jθ)/Π_i=0^n-1 (1 + iθ)

Pr(X = k) = (ⁿ_k) Π_i=0^k-1 (μ + i_θ).Π_j=0^n-k-1 (1 - μ + j_θ)/Π_i=0^n-1 (1 + i_θ), k = 1, 2, 3,.....n.

Question 3:

In an experiment, subjects were exposed to multiple occurrences of the stimuli 'p', 't' and 'k' and, on each occasion, recorded which of these they heard. The results were tabulated as follows.

Propose a probability model for these data using no more than 5 parameters and find maximum likelihood estimators for the parameters.

The following data are reported in the paper Clarke, F R, Constant-ratio rule for confusion matrices in speech communication, Journal of the Acoustical Society of America, 1957. (You do not need to access this paper.)

Fit your model to these data. Does it fit the data well?