Chapter 6 The Normal Probability Distribution PDF

Chapter 6: The Normal Probability Distribution By Ahmad Farooqi, PhD 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 1  Outlines of Chapter 6  C...

Chapter 6: The Normal Probability Distribution By Ahmad Farooqi, PhD 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 1  Outlines of Chapter 6  Continuous Random Variables  Probability Distributions for Continuous Random Variables  The Normal Distribution  Calculation of areas associated with the normal probability distribution  Chapter Review 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 2  Learning Outcomes of Chapter  On completion of this chapter, with the aid of your course textbook, you should be able to know: 1) What is the normal distribution, what is the standard normal distribution. 2) How to compute the probabilities of the standard normal distribution from the standard normal Table (Table 3 in the textbook). 3) How to compute the probabilities of any normal distribution using the standard normal Table. 4) The normal approximation to the Binomial probability distribution: how to compute a Binomial probability by using the normal distribution. 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 3  Introduction  We know a discrete random variables take on only a finite or countably infinite number of values that can be used to model some real-life scenarios.  There are continuous random variables that can assume the infinitely many values corresponding to points on a line interval can also be used to model many real-life scenarios.  The normal probability distribution can be served as models for many practical applications with such kind of continuous random variable. 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 4  Continuous Random Variables  Continuous random variables: Random variables that can assume the infinitely many possible values corresponding to points on a line interval.  Examples:  Heights, weights, Marks, Age can assume any possible values.  Length of life of a particular product.  Experimental laboratory error. 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 5  Probability Histograms  As the number of measurements becomes very large and the class widths become very narrow, the relative frequency histogram appears like a smooth curve, called a probability histogram as shown below. This smooth curve describes the probability distribution of the continuous random variable. 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 6  Continuous Probability Distribution  The depth or density of the probability, which varies with x, may be described by a mathematical formula f(x), called the probability distribution or probability density function for the continuous random variable x.  There are many different types of continuous random variables.  We will briefly introduce the uniform distribution and exponential distribution, then focus on the normal distribution.  We try to pick a model that:  Fits the data well.  Allows us to make the best possible inferences using the data.  Note: A smooth graph of continuous probability distribution is called a curve. 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 7  Area Under the Curve  Note that the Total area under the curve is always equal to (relating to the total probability equal to 1).  Area under the curve between and as shown in blue. In other words, probability of an event between and as shown in blue. 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 8  Properties of Continuous Probability Distributions  If we can calculate the area under the curve, we can solve for the probability of falling within the given arbitrary intervals.  There is no probability attached to any specific value of a continuous random variable. In other words, probability of a continuous r.v at a specific value will always zero, mathematically  That is,. where is any specific value.  This implies that and.  This is not true in general for discrete random variables! 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 9  Continuous Uniform Probability Distribution  The Uniform Random Variable is used to model the behavior of a continuous random variable whose values are uniformly or evenly distributed over a given interval.  If takes on values between and , then probability density function for uniform distribution is given by  This results in an area under the curve of 1, which is easy to confirm as it produces a rectangle with width and height ,  Note: The values of a uniform random variable are evenly distributed over a given interval. 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 10  Uniform Probability Distribution-Example  Example (Textbook) Rounding Error: The error introduced by rounding an observation to the nearest centimeter has a Uniform Distribution over the interval.  What is the probability that the rounding error is less than 0.2?  Solution: 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 11  Exponential Probability Distribution  The Exponential Random Variable is used to model continuous random variables such as waiting times, or more importantly, lifetimes (life length) associated with electronic components.  In general, the Exponential probability density function is given by /  Parameter is the mean of the distribution.  It can be shown that / for.  Memoryless Property: A unique property of the exponential distribution is 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 12  Exponential Probability Distribution-Example  Example 6.1 (Textbook) Waiting Time: The waiting time at a Canadian supermarket checkout has an exponential distribution with an average time of five minutes. The probability density function is visualized below.  What is the probability that you need to wait 10 minutes or more?  Solution:  9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 13  The Normal Random Variable  Recall a continuous random variable, then the probability distribution discovered by Gauss in 1890, based on continuous r.v.  Normal Random Variable: A random variable, X is said to be a normal random variable, with mean μ and standard deviation σ, if it can assume all possible values in an interval [-∞, ∞]. For example, the height of a person, the temperature of the place, etc. 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 14  The Normal Probability Distribution  Let a continuous random variable X follows a bell-shaped curve called a normal distribution with mean (μ) and SD (σ), then its probability function is defined as: ( ) for Where, μ= Mean of Normal distribution, σ= SD of Normal distribution ,  A typical normal curve looks like this:  We indicate that a random variable X is normally distributed as X N(μ, ). 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 15  Characteristics of Normal Probability Distribution  A normal distribution is a bell-shaped curve.  A range of normal curve is from.  The mean and SD of normal distribution is μ and σ respectively. μ  A normal curve is symmetric about the mean μ.  The area under an entire normal curve is.  The height of any normal curve is maximized at x = µ.  The shape of any normal curve depends on its mean μ and SD σ.  Increasing the mean shifts the density curve to the right...  Increasing the SD flattens the density curve... 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 16 Characteristics of Normal Probability Distribution (can’t)  Since a normal curve is symmetrical about the mean therefore, Mean=Median=Mode  The 68% of the data fall between μ  The 95% of the data fall between μ  The 99% of the data fall between μ  The first and third quartiles are  The 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 17  The Area in Normal Probability Distribution  The probability density function for a normal distribution with mean and standard deviation looks like with of area above the mean and of area below the mean : 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 18  The Location and Shape of Normal Distribution  The shape and location of the normal curve changes as the mean and standard deviation change.  The mean locates the centre of the distribution.  The shape of the distribution is determined by , the population standard deviation.  Large values of reduce the height of the curve and increase the spread.  Small values of increase the height of the curve and reduce the spread. 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 19  Standard Normal Random Variable  Standard Normal Random variable: A random variable, z=(( −μ)/σ ) is said to be a Standard Normal Random variable (z score), with mean μ and SD σ. The probability distribution of z is called a Standard Normal distribution.  To find ( < < ), we need to find the area under the appropriate normal curve.  To simplify the tabulation of these areas, we standardize each value of by expressing it as a z-score, the number of standard deviations it lies from the mean :  Equivalently, = +z.  Note: 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 20  Characteristics of Standard Normal Random Variable  Mean ; Standard deviation  When  Symmetric about  Values of z to the left of Centre are negative  Values of z to the right of Centre are positive  Total area under the curve is 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 21  The Area in Standardized Normal Distribution  Random variable z has a normal distribution with and.  We can solve for cumulative probabilities using Table 3 (page: 720-721 textbook). The shaded area represents. 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 22  Finding Area Using Table 3  The 4-digit probability in a particular row and column of Table 3 (page:720-721 textbook) gives the area under the z curve to the left that specific value of.  What is ? Area for z0 = 1.00 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 23  Finding Area Using Table 3-Example  A probability of the form can be obtained from Table 3 (page:720-721 textbook) directly. 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 24  Finding Area Using Table 3-Example  A probability of the form can be calculated based on the complement probability.  To find an area to the right of a z-value, find the area in Table 3 (page:720-721 textbook) and subtract from 1. 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 25  Finding Area Using Table 3-Example  Like with our discrete distributions, a probability of the form can be calculated from two cumulative probabilities.  To find an area between two values of , find the two areas in Table 3, and subtract one from the other. 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 26  Finding The Value of z Using Table 3-Example  One can use Table 3 (page:720-721 textbook) to solve for the value of that results in a given area or probability.  Example Working Backwards 1: Find the value of such that  Solution: 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 27  Finding The Value of z Using Table 3-Example  Again, one can use Table 3 (page:720-721 textbook) to solve for the value of that results in a given area or probability.  Example Working Backwards 2: Find the value of such that..  Solution: 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 28  Finding The Value of z Using Table 3-Example  Example Working Backwards 3: Find the value of such that 0.95 of the area is within standard deviations of the mean.  Solution:  By symmetry of the standard normal distribution, there must be an area of. to the right of and to the left of , for.  Thus, the area to the left of equals  This number is found in the interior of Table 3 in the row for and the column.  So,. 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 29  The Empirical Rule  Recall: By the Empirical Rule, approximately of measurements lie within standard deviations of the mean.  We have now shown that there is exactly within standard deviations of the mean, which rounds to. 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 30 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 31 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 32  Finding Area (Probability) Using Table 3-Exercise  Exercise: Use Table 3 (page:720-721 textbook) to compute area or probabilities with given z-values for a standard normal random variable. 1. =? 2. =? 3. =? 4. =? 5. =? 6. =? 7. =? 8. =? 9. =? 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 33 Finding Area (Probability) Using Table 3-Exercise (cont’d) 1. Since the standard normal Table gives only areas to the left of a given z values, we can re-write this probability as the difference between two probabilities, each involving area to the left of a number (see also portions of the Table indicating this computations): (1.24 < z < 1.70) = (z < 1.70) − (z < 1.24) = 0.9554-0.8925 = 0.0629 9/26/2024 Chapter 6: The Normal Probability Distribution by Ahmad Farooqi, PhD 34 Finding Area (Probability) Using Table 3-Exercise (cont’d) 8. This area is to the left of a number and so we don't have to manipulate. However, the number -0.437 falls between -0.43 and -0.44 but it is close to -0.44. Therefore, you have two options, either find the areas P(z< -0.44) and P(z