4 - Probability distributions

Summary

Every measurement is in fact a random sample from a probability distribution. In order to make a judgment on the accuracy of an experimental result we must know something about the underlying probability distribution. This chapter treats the properties of probability distributions and gives details about the most common distributions. The most important distribution of all is the normal distribution, not in the least because the central limit theorem tells us that it is the limiting distribution for the sum of many random disturbances.

Introduction

Every measurement x_i of a quantity x can be considered to be a random sample from a probability distribution p(x) of x. In order to be able to analyze random deviations in measured quantities we must know something about the underlying probability distribution, from which the measurement is supposed to be a random sample.

If x can only assume discrete values x = k, k = 1, …, n then p(k) forms a discrete probability distribution and p(k) (often called the probability mass function, pmf) indicates the probability that an arbitrary sample has the value k. If x is a continuous variable, then p(x) is a continuous function of x: the probability density function, pdf. The meaning of p(x) is: the probability that a sample x_i occurs in the interval (x, x + dx) equals p(x) dx.