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Chapter 10 More Continuous Probability Models 10.1 Introduction Over the past few weeks we have discussed some “standard” probability distributions which can be used to model data. We have looked at two such distributions for discrete data – the binomial distribution and the Poisson distribution – and last week the Normal distribution was introduced as a probability model for continuous data. Recall the probability density function (pdf) of the Normal distribution, which is often referred to as a “bell–shaped curve”: x f ( x ) μ μ - 2 σ μ +2 σ μ - 4 σ μ +4 σ We saw in the lecture last week that many naturally occurring continuous measurements (such as height, weight, time, rainfall etc.) often resemble this bell–shaped curve when plotted using a histogram, for example. But what if we cannot assume “Normality” for our data? We now consider two other probability models which can be used to model continuous data when the Normal distribution isn’t appropriate. 98
CHAPTER 10. MORE CONTINUOUS PROBABILITY MODELS 99 10.2 The Uniform Distribution The uniform distribution is the most simple continuous distribution. As the name suggests, it describes a variable for which all possible outcomes are equally likely. For example, suppose you manage a group of Environmental Health Officers and need to decide at what time they should inspect a local hotel. You decide that any time during the working day (9.00 to 18.00) is okay but you want to decide the time “randomly”. Here “randomly” is a short–hand for “a random time, where all times in the working day are equally likely to be chosen”. Let X be the time to their arrival at the hotel measured in terms of minutes from the start of the day. Then X is a uniform random variable between 0 and 540 : As with the Normal distribution, the total area (base × height) under the pdf must equal one. Therefore, as the base is 540 , the height must be 1 / 540 . Hence the probability density function (pdf) for the continuous random variable X is f ( x )= 1 540 for 0 x 540 0 otherwise . In general, we say that a random variable X which is equally likely to take any value between a and b has a uniform distribution on the interval a to b , i.e. X U ( a, b ) . The random variable has probability density function (pdf) f ( x )=
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