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Unformatted text preview: LECTURE  8 Continuous Probability Distributions Normal Probability Distribution Read: Chapter 6, Section  6.2. The normal probability distribution is a continuous probability distribution that is widely used in a variety of applications. It has the following characteristics: 1. It is typically bell shaped in form. 2. It is symmetric around the mean  the shape of the curve to the left of the mean being a mirror image of the curve's shape to the right of the mean. Note: The mean line is the vertical line at the center of the curve. 3. The curve's highest point is at the mean. 4. The mean = median = mode. 5. The area under the curve = 1. As the curve is symmetric around the mean, therefore the mean line divides the curve into two halves, each having an area of 0.5 . 6. The normal random variable assumes a value within ± 1 σ of the mean about 68% of the time; within ± 2 σ of the mean about 95% of the time; and within ± 3 σ about 99% of the time. 7. The ends of the curve are at ± ∞ . The parameters to describe a normal probability distribution are μ and σ . To show that a random variable x is normally distributed, we write x ~ N( μ , σ ) 1 The Standard Normal Probability Distribution This special case of a normal distribution has a mean ( μ ) = 0 and standard deviation ( σ ) = 1. We denote the standard normal random variable as z and write z ~ N(0,1). Note:The normal random variable can take any value in some interval. To compute the probability that a normal random variable is within some interval the area under the standard normal curve has to be computed. These probabilities have been computed and provided in tables that are available in any statistics book. (In your textbook, the areas or probabilities for the standard normal table are provided in the inside front cover ). 2 EXAMPLE 1 We want to know the probability that the standard normal random variable z is less than 0.6. This means that z lies in the interval ( ∞ to 0.6). This can be written as P( ∞ ≤ z ≤ 0.6) or P( z ≤ 0.6) We need the area under the standard normal curve between z =  ∞ and z = any positive number (in our case z = 0.6). Looking at the standard normal table, we look at the first column and locate z = 0.6 . Then we travel right in the row of z = 0.6 and locate the entry under the column .00. This is .7257 . Therefore the probability that z lies in the interval ( ∞ to 0.6); i.e., P( z ≤ 0.6) = 0.7257 or 72.57%. Similarly we can compute the probabilities of z assuming any value between  ∞ and a positive number or z being less than a positive number. 3 Important Note: Microsoft Excel can be used to solve the Example 1. Excel computes the cumulative probability. Therefore it will give a solution that will provide a probability value of z lying in the interval ( ∞ to 0.6)....
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 Summer '08
 SINGER

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