The number of times that a person contracts a cold in a...

The number of times that a person contracts a cold in a given year is a Poisson random variable with parameter $\lambda=5 .$ Suppose that a new wonder drug (based on large quantities of vitamin $\mathrm{C}$ ) has just been marketed that reduces the Poisson parameter to $\lambda=3$ for 75 percent of the population. For the other 25 percent of the population, the drug has no appreciable effect on colds. If an individual tries the drug for a year and has 2 colds in that time, how likely is it that the drug is beneficial for him or her?

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Key Concepts

Poisson Distribution

This is a discrete probability distribution used to model the number of times an event occurs in a fixed interval of time or space when the events occur independently at a constant average rate. It is characterized by its parameter ?, which represents the mean number of occurrences in the interval.

Bayesian Inference

Bayesian inference is a method of statistical analysis that updates the probability for a hypothesis as more evidence or information becomes available. It uses Bayes’ theorem to combine prior probability (what was believed before the data) with a likelihood function (the probability of the data under various hypotheses) to produce a posterior probability.

Mixture Model

A mixture model is used when data is generated from several different subpopulations, each with its own probability distribution. In such cases, the overall probability distribution is a weighted sum of the individual distributions, with the weights representing the proportion of the population corresponding to each subpopulation.

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