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notes515fall10chap5 - STAT 515 - Chapter 5: Continuous...

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STAT 515 -- Chapter 5: Continuous Distributions Probability distributions are used a bit differently for continuous r.v.’s than for discrete r.v.’s. Continuous distributions typically are represented by a probability density function (pdf), or “density curve.” (kind of a “theoretical” histogram) A density curve is a representation of the underlying population distribution (not a description of actual sample data). The normal distribution is a particular type of continuous distribution. Its density has a bell shape: Properties of Density Functions: (1) Density function always on or above the horizontal axis (curve can never have a negative value) (2) Total area beneath the curve (between curve and horizontal axis) is exactly 1. (3) An area under a density function represents a probability about the r.v. (or the proportion of observations we expect to have certain values).
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With discrete r.v.’s we looked at probability function (table, graph) to find probability of the r.v. taking a particular value. For continuous r.v.’s, the probability distribution will give us the probability that a value falls in an interval (for example, between two numbers). That is, the probability distribution of a continuous r.v. X will tell us P( a X b ), where a and b are particular numbers. Specifically, P( a X b ) is the area under the density function between x = a and x = b . Examples:
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The Uniform Distribution This is a simple example of a continuous distribution. A uniform r.v. is equally likely to take any value between its lower limit (some number c ) and its upper limit (some number d ). Density looks like a rectangle:
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notes515fall10chap5 - STAT 515 - Chapter 5: Continuous...

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