Chapter 6 Continuous Random Variables

Chapter 6Continuous Random VariablesCopyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/IrwinContinuous Random Variables6.1 Continuous Probability Distributions6.2 The Uniform Distribution6.3 The Normal Probability Distribution6.4 Approximating the Binomial Distribution by Using the Normal Distribution (Optional)6.5 The Exponential Distribution (Optional)6.6 The Normal Probability Plot (Optional)6-*6.1 Continuous Probability DistributionsA continuous random variable may assume any numerical value in one or more intervalsFor example, time spent waiting in lineUse a continuous probability distribution to assign probabilities to intervals of values The curve f(x) is the continuous probability distribution of the random variable x if the probability that x will be in a specified interval of numbers is the area under the curve f(x) corresponding to the intervalLO6-1: Define a continuous probability distribution and explain how it is used.6-*Properties of Continuous Probability DistributionsProperties of f(x): f(x) is a continuous function such thatf(x) ≥ 0 for all xThe total area under the f(x) curve is equal to 1Essential point: An area under a continuous probability distribution is a probabilityLO6-16-*6.2 The Uniform DistributionIf c and d are numbers on the real line (c < d), the probability curve describing the uniform distribution isThe probability that x is any value between the values a and b (a < b) isNote: The number ordering is c < a < b < dLO6-2: Use the uniform distribution to computeprobabilities.6-*6.3 The Normal Probability DistributionThe normal probability distribution is defined by the equationfor all values x on the real number line σ is the mean and σ is the standard deviation π = 3.14159 and e = 2.71828 is the base of natural logarithmsLO6-3: Describe the properties of the normal distribution and use a cumulative normal table.6-*The Standard Normal TableThe standard normal table is a table that lists the area under the standard normal curve to the right of negative infinity up to the z value of interestTable 6.1Other standard normal tables will display the area between the mean of zero and the z value of interestAlways look at the accompanying figure for guidance on how to use the tableLO6-36-*Find P(0 ≤ z ≤ 1)Find the area listed in the table corresponding to a z value of 1.00Starting from the top of the far left column, go down to “1.0”Read across the row z = 1.0 until under the column headed by “.00”The area is in the cell that is the intersection of this row with this columnAs listed in the table, the area is 0.8413, so P(– ≤ z ≤ 1) = 0.8413P(0 ≤ z ≤ 1) = P(– ≤ z ≤ 1) – 0.5000 = 0.3413LO6-4: Use the normal distribution to compute probabilities.6-*Finding Normal ProbabilitiesFormulate the problem in terms of x valuesCalculate the corresponding z values, and restate the problem in terms of these z valuesFind the required areas under the standard normal curve by using the tableNote: It is always useful to draw a picture showing the required areas before using the normal tableLO6-5: Find population values that correspond to specified normal distribution probabilities.6-*Some Areas under the Standard Normal CurveLO6-5Figure 6.156-*Finding a Tolerance Interval Finding a tolerance interval [  k] that contains 99% of the measurements in a normal populationLO6-5Figure 6.236-*6.4 Approximating the Binomial Distribution by Using the Normal Distribution (Optional)The figure below shows several binomial distributionsCan see that as n gets larger and as p gets closer to 0.5, the graph of the binomial distribution tends to have the symmetrical, bell-shaped, form of the normal curveLO6-6: Use the normal distribution to approximate binomial probabilities (Optional).Figure 6.246-*Normal Approximation to the Binomial ContinuedGeneralize observation from last slide for large pSuppose x is a binomial random variable, where n is the number of trials, each having a probability of success pThen the probability of failure is 1 – pIf n and p are such that np  5 and n(1–p)  5, then x is approximately normal withLO6-66-*6.5 The Exponential Distribution (Optional)Suppose that some event occurs as a Poisson processThat is, the number of times an event occurs is a Poisson random variableLet x be the random variable of the interval between successive occurrences of the eventThe interval can be some unit of time or spaceThen x is described by the exponential distributionWith parameter , which is the mean number of events that can occur per given intervalLO6-7: Use the exponential distribution to compute probabilities (Optional).6-*6.6 The Normal Probability PlotA graphic used to visually check to see if sample data comes from a normal distributionA straight line indicates a normal distributionThe more curved the line, the less normal the data isLO6-8: Use a normal probability plot to help decide whether data come from a normal distribution (Optional).6-*