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## Chapter 6 Continuous Random Variables

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**Chapter 6 Continuous Random Variables**• Continuous Probability Distributions • The Uniform Distribution • The Normal Probability Distribution Dr. Constance Lightner- Fayetteville State University**Continuous Probability Distributions**• A continuous random variable can assume any value in an interval on the real line or in a collection of intervals. • The function f(x) is the probability density function (or probability distribution function) of the continuous random variable x. • Unlike a discrete random variable, we can not simply plug values of the random variable into this function and get probability information directly. • For continuous random variables it is impossible to talk about the probability of the random variable assuming a particular value. • Instead, we talk about the probability of the random variable assuming a value within a given interval. • In order to determine the probability that a continuous random variable assumes a value in an interval you must first draw the function f(x). Dr. Constance Lightner- Fayetteville State University**Properties of a Continuous Random Variable**• The probability density function f(x)0 for all values of x. • The probability of a continuous random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1and x2. • The probability of a continuous random variable assuming a specific value is zero (there is no area under any graph at an exact point). • The total area under the graph of f(x) equals 1. Dr. Constance Lightner- Fayetteville State University**How to find the Probability that a Continuous Random**Variable Falls within an Interval? In order to find the probability that a continuous random variable, x falls in an interval between x1 and x2 do the following: • Graph the probability density function. • Identify the interval of interest on the x axis. • Shade in the area under f(x) in this interval. • Compute the area of the shaded region. The area of the shaded region is the probability that x will fall between x1 and x2 . The probability that x falls in the interval x1 to x2 is the same as the proportion of x values from the population that fall between x1 and x2 . Dr. Constance Lightner- Fayetteville State University**Special Random Variables**In this chapter we will discuss two popular continuous random variables: • Uniform random variable • Normal random variable Dr. Constance Lightner- Fayetteville State University**Uniform Probability Distribution**• A random variable is uniformly distributed whenever it is equally likely that a random variable could take on any value between c and d. • The uniform random variable has the following probability density function: f(x) = 1/(b - a) for a <x< b = 0 elsewhere where: a = smallest value the variable can assume b = largest value the variable can assume Dr. Constance Lightner- Fayetteville State University**Uniform Probability Distribution**• Expected Value of x E(x) = (a + b)/2 • Variance of x Var(x) = (b - a)2/12 Dr. Constance Lightner- Fayetteville State University**Example 6.2**The service time at a CAL’s restaurant is uniformly distributed between 5 and 15 minutes. The probability density function is f(x) = 1/10 for 5 <x< 15 = 0 elsewhere where: x = the service time for a customer Dr. Constance Lightner- Fayetteville State University**What is the probability that the time that it will take to**service a customer is between 12 to 15 minutes ? Dr. Constance Lightner- Fayetteville State University**Width= 3**Height=0.1 Dr. Constance Lightner- Fayetteville State University**The area under f(x) between 12 and 15 is a rectangle.**area= 3* 0.1=0.3 Thus the probability that it takes between 12 to 15 minutes for a customer to get serviced is 0.3. Moreover, we can conclude that 30% of the CAL’s customers will wait between 12 to 15 minutes for service. Dr. Constance Lightner- Fayetteville State University**What is the probability that the time that it will take to**service a customer is between 7 to 12 minutes ? =5* 0.1= 0.5**Example 6.2(Expected Value and Variance)**• Expected Service time = (5 + 15)/2 = 10 minutes • Variance of Service times =(15-5)2 /12 = 8.33 minutes Dr. Constance Lightner- Fayetteville State University**Normal Probability Distribution**• The normal probability distribution is the most popular and important distribution for describing a continuous random variable. • This distribution has been used to define: • This distribution is used in various statistical inference techniques • Heights and weights of people • Test scores • IQ Scores • The Normal distribution is widely used in various statistical inference techniques. Dr. Constance Lightner- Fayetteville State University**Normal Probability Distribution**The probability density function for a normal random variable is: where: = mean = standard deviation = 3.14159 e = 2.71828 Dr. Constance Lightner- Fayetteville State University**Characteristics of the Normal Probability Distribution**• The distribution is symmetric, and illustrated as a bell-shaped curve. • Two parameters, (mean) and (standard deviation), determine the location and shape of the distribution. • The highest point on the normal curve is at the mean, which is also the median and mode. • The mean can be any numerical value: negative, zero, or positive. Dr. Constance Lightner- Fayetteville State University**The total area under the curve is 1 (.5 to the left of the**mean and .5 to the right). Bowerman, et. al Dr. Constance Lightner- Fayetteville State University**Probabilities for the normal random variable are given by**areas under the curve. Bowerman, et. al**The Position and Shape of the Normal Curve**Bowerman, et. al**68.26% of values of a normal random variable are within +/-**1standard deviation of its mean. • 95.44% of values of a normal random variable are within +/- 2standard deviations of its mean. • 99.73% of values of a normal random variable are within +/- 3standard deviations of its mean. Bowerman, et. al**In order to better understand the normal**probability distribution you should STOP here, go to Course Documents, click on Chapter 6, and complete the Normal Distribution Exercise. Check your answers, and then return to Chapter 6 part 2 notes. Dr. Constance Lightner- Fayetteville State University