Chapter5 - Chapter 5 Continuous Probability Distribution...

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Unformatted text preview: Chapter 5: Continuous Probability Distribution Chapter 5: Continuous Probability Distribution Keith E. Emmert Department of Mathematics Tarleton State University June 16, 2011 Chapter 5: Continuous Probability Distribution Outline 1 Why Do We Need to Know Models? 2 Modeling Continuous Variables 3 Normal Distributions Chapter 5: Continuous Probability Distribution Why Do We Need to Know Models? Example Scenario I: A patient visits his doctor complaining of a number of symptoms. The doctor suspects the patient is suffering from some disease. The doctor performs a diagnostic test to check for this disease. High responses on the test support that the patient may have the disease. The patients test response is 200. What does this say? 80 100 120 140 160 180 200 Model for healthy subjects Test Response Based on this model, it is very unlikely that a test response of 200, or greater, would have occurred if the subject were actually healthy. Thus, either the patient has this disease or a very unlikely event has occurred. Chapter 5: Continuous Probability Distribution Why Do We Need to Know Models? Example Scenario II: Suppose we wish to compare two drugs, Drug A and Drug B, for relieving arthritis pain. Subjects suitable for the study are randomized to one of the two drug groups and are given instructions for dosage and how to measure their “time to relief.” Results of the study are summarized by presenting the models for the time to relief for the two drugs. Time to relief Drug B Drug A t Consider any point in time, say time t as indicated on the above axis. A higher proportion of subjects treated with Drug A have felt “relief” by this time point as compared to those treated with Drug B. If the study design was sound, then we might conclude that Drug A works quicker than Drug B. Chapter 5: Continuous Probability Distribution Modeling Continuous Variables Density Functions A density function is a (nonnegative) function or curve that describes the overall shape of a distribution. The total area under the entire curve is equal to 1, and proportions or probabilities are measured as areas under the density function. As a simple example, below is a density curve (the blue curve). The shaded area represents the probability that a random variable takes on values between 6 and 20. 6 20 Chapter 5: Continuous Probability Distribution Modeling Continuous Variables Let’s Do It! Using Density Functions Let the variable X represent the length of life, in years, for an electrical component. The following figure is the density curve for the distribution of X . 1 2 3 4 5 6 x Density 0.39 0.24 0.14 0.09 0.05 0.03 (a) What proportion of electrical components lasts longer than 6 years?...
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Chapter5 - Chapter 5 Continuous Probability Distribution...

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