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Unformatted text preview: Chapter 6 Continuous Distributions A. Introductory Information-the normal curve, uniform, and exponential are part of a family of distributions that we call continuous probability distributions. 1. Continuous Probability Distribution the values that the data takes on is now continuous. The probability of any single point is 0. We find the probability of some event occurring given an interval. 2. Probability Density Function this is the actual distribution we deal with. It is specific to a particular type of distribution. We denote it f(x); this is the same thing that is called a density curve. The function just summarizes the curve. Properties: (a) the total area under the curve is 100% (b) the total range of values is on the horizontal axis (c) the median still gives us the 50% point and the mean defines where the balance point of the curve would be (i.e. think of it as the middle of the probability density) Examples: B. Types of Distributions 1. Uniform Distribution every unit interval is equally likely f(x) = 1 / ( b a ) for all a x b where a = small value b = large value 0 elsewhere Graphically: 1 x x f (x) the density function defines the distribution a b x Properties (a) the area under the curve is 1 or 100% (b) there is only probability when we talk about some given interval; it is the area under the curve in the given interval. If we didnt have an interval the area is 0 at one particular point. (c) P (a x b) = x * f (x) (d) i-Mean - E (x) = (a + b) / 2 ii- Variance - Var (x) = (b a) 2 / 12 *** these are the variances and average of the distribution. E (x) is called the expected value. It essentially tells you the mean of a distribution Ex: Given that a = 2 and b = 8 find P (1 x 5), E(x) and Var (x). We find it with the above formulas. First we note that the distribution is only defined over 28, so we can note that P (1 x 2) = 0 and we can re-write our interval of interest as P (2 x 5). We then note that f(x) = 1/6 in the interval from 2 to 8. The interval x is 5-2=3. So 3 * f(x) = 3 (1/6) = .5 Graphically: E(x) = (8+2)/2 = 5; so the center of the mean of the distribution is 5. Var (x) = 36/12= 3 2. Normal Probability Curve this is the most widely used distribution; any distribution with a bell shape is considered a normal distribution....
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- Spring '10