Probability Distributions: Mutations, Cancer Rates, and Vision Sensitivity

Abstract

In this chapter reviews the rules of probability. Then, we introduce the important concept of a random variable and the equally important concept of a probability distribution and express the mean, variance, and standard deviation of a random variable in terms of its probability distribution. We then focus on a certain, very-widely applicable probability distribution, namely the Poisson distribution, which specifies the probability of counting a certain number of occurrences or events in a given region or interval under many circumstances. We then examine several applications of the Poisson distribution, including its application to counting the number of bacterial colonies on a Petri dish and its application to the incidence of retinoblastoma, a childhood eye cancer. An understanding of probability is indispensable for properly designing appropriate experiments to test hypotheses and for evaluating the experimental evidence so-obtained. Whenever an experiment involves counting, the Poisson distribution is more than likely applicable. We also introduce another important distribution, namely the binomial distribution. We then discuss the mean and variance of a composite, ”big” random variable, that is formed as the sum of several ”small” random variables. Importantly, we will see that the mean of such a ”big” random variable is the sum of the means of the ”small” random variables, and that its variance is the sum of the variances of the ”small” random variables. We finish up by introducing the exponential and Gaussian distributions, and the ”central limit theorem”, which states that a composite random variable has a Gaussian distribution, provided its component random variables are numerous enough.