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Unformatted text preview: Chapter Four Continuous Random Variables and Probability Distributions 4.1 Continuous Random Variables and Probability Density Functions For a discrete random variable X , one can assign a probability to each value that X can take (i.e., through the probability mass function). Random variables such as heights, weights, lifetimes, and measurement error can all assume an infinite number of values. As a result, we need a different mechanism for understanding the probability distribution. For a continuous random variable , any real number x is possible between A and B (for A < B ). The number of values is theoretically infinite, but one wont have such precision in practice. Repre- senting such a random variable using a continuous model is still appropriate. Note that a continuous random variable can still have specific endpoints to its range (e.g., x > 0, 0 < x < 1 or x > 1), but the number of values possible is still infinite. As discussed earlier, we can use histograms to represent the relative frequency of a random variable X . Suppose X is the depth of a lake at a randomly chosen point on the surface. X has a range from 0 to M , where M is the maximum depth of the lake. The relative frequency histogram can represent the probability distribution for X . The finer the discretization of the X axis (i.e., the precision of the measurement), the smaller the subintervals for the histogram as seen below. Infinitely small subintervals lead to the continuous probability distribution being represented as a smooth curve. As with the relative frequency histograms, the probability associated with the values of X between any two values a and b is the area under the smooth curve between a and b . For functions consisting of straight line segments, geometry and simple relationships for known areas (shapes) can be used. For more complex functions, we may be able to subdivide the area under the curve into smaller rectangles and sum their areas. However, calculus and integration techniques solve this for us. If X is a continuous random variable, the probability distribution or the probability density function ( pdf ) of X is a function f ( x ) such that for any two numbers a and b with a b : P ( a X b ) = Z b a f ( x ) dx. This probability is the area under the curve f ( x ) between the points a and b . For f ( x ) to be a legitimate density function, the following two conditions must hold: f ( x ) 0 for all x , and 1 R - f ( x ) dx = 1. With discrete random variables, we only had a finite set of values for X , each with probability mass at x and the sum of possible p ( x ) values equal to 1 (the equivalent of the two conditions above)....
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- Fall '10