Edge(X) Notes: Module 3 Random Variables Random Variables: - Sample Space (Omega): for a random process is the set of all possible outcomes that might be observed. - A probability function assigns a probability to every subset (event) of the sample space. - These probabilities must all be between 0 and 1, and the probability of omega, the whole sample must be 1. - A random variable is a random process with a numerical outcome - Random variables can be discrete or continuous. Discrete random variables typically arise from counting. Continuous random variables arise from measurements, such as picking a student at random and recording their height. - Remember that capital letters denote a random process while lowercase letters denote observations. Population and Sampling: - Instead we take a sample, a subset of the population, and use data from the sample to say something about the population. - One way of achieving this is to take a random sample from the population. - A parameter is a numerical characteristic of a population. - The symbol 'u' is the Greek letter 'm', as in 'mean'. We will almost always use lowercase Greek letters to denote population parameters. Continuous Random Variables: - A continuous quantity is one that can be measured to arbitrary precision. - For example, it makes sense to talk about the probability of a height being between 160.0 cm and 160.1 cm. A probability function for a continuous random variable is called a probability density function (pdf) and is plotted as a density curve. - The pdf must always be positive (or zero) and the total area under the density curve must be 1. The probability of an interval event is simply the area under the density curve above that interval. - Our aim for continuous random variables is to provide models for experiments involving continuous variables, such as height measurements. Conditional Probability
Conditional Probability: - We use P(A) to denote the probability of an event A occurring, this tells us how likely the event is to occur. - We use P(A| B) to denote the conditional probability of A occurring if B has occurred. - Conditional probabilities are calculated using the rule; P(A | B) = P (A and B) P(B) - We say that two events, A and B, are independent if the outcome of one tells us nothing about the probability distribution of the other. That is, P(A|B) = P(A) and P(B|A) = P(B). In most cases you can decide whether two events are independent by thinking about the physical way in which they happen. If events are independent then the multiplication rule in the previous section becomes P(A and B)=P(A)P(B). Example Question 1 The first randomised response methodology was presented by Warner (1965) and was based on a different process to the example in this section. For example, suppose 200 university students are asked "If you or your girlfriend accidentally got pregnant would you seriously consider the possibility of an abortion?". Before answering, each student