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Unformatted text preview: CHAPTER 3 CONTINUOUS RANDOM VARIABLES AND THE NORMAL DISTRIBUTION 1. Difference Between Discrete and Continuous Random Variables 1.1. Discrete Random Variables 1.2. Continuous Random Variables 2. The Normal Distribution 2.1. Finding Probabilities in a Normal Distribution 2.1.1. Using the Standard Normal Distribution to Find Probability of x 2.1.2. The Standard Normal Table (zTable) 2.2. Finding the x Value for a Given Probability 2.3. z scores Frequently Used in Inferential Statistics Random variable are divided into two general categories: discrete random variable and continuous random variables. 1. Difference Between Discrete and Continuous Random Variables The difference between the probability distribution of discrete versus continuous random variables can be explained by the way they are presented graphically. 1.1.Discrete Random Variables In the case of guessing the answers to the 5-question multiple choice exam and letting x denote the number of correct answers guessed, the probability distribution of x is as follows: x f ( x ) 0.2373 1 0.3955 2 0.2637 3 0.0879 4 0.0146 5 0.0010 1.0000 Since x is discrete, it takes on discrete values and the probability of each value is specified. If you can enumerate or list every value of the random variable, and each value has a non-zero probability, then you have a discrete random variable. Note: Some textbooks use the notation P( x ) in place of f ( x ). The term f ( x ) is called a probability density function . Probability density refers to the point on the graph corresponding to a given value of the random variable x . In other words, probability density is the height of the graph at the given value of the random variable. In the chart below, for example, f (2) = 0.2637. Chapter 3—The Normal Distribution Page 1 0.2373 0.3955 0.2637 0.0879 0.0146 0.0010 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 1 2 3 4 5 f ( x ) x Probability Density Function of a Discrete Random Variable 1.2.Continuous Random Variables Unlike a discrete random variable, you cannot enumerate or list all the values of a continuous random variable. A continuous random variable takes on infinite number of values within an interval. Graphically, the distinguishing features of a continuous random variable are: The density function of a continuous random variable, f ( x ), is not represented by a bar graph. It is, rather, shown as a continuous graph, a smooth curve. Because probability cannot be defined for a single value of x , the height of f ( x ) at a given value of x does not represent the probability of that value. The continuous random variable can take on any of the infinite number of values in a given range. Thus the probability that x will be equal to a single value is 1/∞ , which is 0....
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- Fall '10
- Normal Distribution, Xu