Continuous distributions
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Continuous distributions. Probability density function
That is, for a continuous function, f(x), if [Equation goes here &- download the original to see it.] And provided [Equation.]1 for all x, then f (x) defines a probability distribution. The function f(x) is called a continuous probability density function. The condition [Equation.] for all x is necessary because we cannot have negative probabilities. Example [Equation.] The function is represented by a straight line. The area of the triangle under the line is 1. [Diagram.] [Equation.] This example also illustrates that we should be precise about the definition of the probability density function. [Equation.] We should state explicitly what the values of f(x) are outside this domain. [Equation.] Thus, our continuous probability density function f(x) is really the piecewise addition of three parts. [Equation.] Because f(x) has been constructed from "pieces", we say it has been defined piecewise. We write the function as&: [Equation.] Clearly, probability density functions can be constructed from all sorts of pieces &- each piece specified by a different function. This is illustrated by the following example. Example Show that the function [Equation.] is a probability density function, and sketch its curve. [Equation.] Since f(x)>0 for all x, f(x) is a p.d.f. (abbreviation for probability density function). It looks like this [Diagram.] Sometimes questions are set that ask you to find the form of a probability density function, or rather, a parameter as in the following example. Example A continuous random variable X has probability density function [Equation.] Find the value of k. [Equation.]
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Contents of
Continuous distributions
1 Continuous random variables
2 Continuous distributions. Probability density function
3 Expectation of a continuous probability distribution
4 Variance of a continuous probability distribution.
5 Expectation of a general function of a random variable X
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