Chapter 6 Continuous Random Variables

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

Chapter 6 Continuous Random Variables