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Unformatted text preview: Chapter 3: Continuous Random Variables 3. The Continuous Sample Space Continuous random variables range over continuous sets of numbers, which are sometimes referred to as intervals . An interval contains all of the real numbers between two limits. For the limits x 1 and x 2 with x 1 < x 2 , there are four different intervals distinguished by which of the limits are contained in the interval: • ( x 1 ,x 2 ) = { x  x 1 < x < x 2 } . • [ x 1 ,x 2 ] = { x  x 1 ≤ x ≤ x 2 } . • [ x 1 ,x 2 ) = { x  x 1 ≤ x < x 2 } . • ( x 1 ,x 2 ] = { x  x 1 < x ≤ x 2 } . Many experiments lead to random variables with a range that is a continuous interval. Some examples include measuring the arrival time of a shipment, measuring the voltage across a resistor, measuring the Ph of a chemical compound, and measuring the height of an athlete’s vertical leap. Again, the axioms of probability dictate that we assign numbers between zero and one to all the probabilities of events (i.e., sets of elements) in the sample space. A random variable X is said to be continuous if its set of possible values constitutes an entire interval of numbers. However, there is a unique aspect to models of continuous random variables. Because there are an infinite number of points or values between any two values on the interval, the probability that the random variable assumes a value corresponding to exactly one of those points is zero. Thus, even if the event X = x is possible , it has a probability of zero . Consequently, when working with continuous random variables our concern will be with computing probabilities for various intervals, such as P [ X > a ] or P [ a < X < b ]. Given this perspective to assigning probabilities to continuous random variables, it follows that: P [ a < X ≤ b ] = P [ a < X < b ] = P [ a ≤ X < b ] = P [ a ≤ X ≤ b ] . In general, if the scale used to measure a random variable X can be subdivided to any extent desired (at least in concept), then the variable is continuous; if it cannot, the variable is discrete. A frequently encountered probability model is that of a uniform random variable. The sample space of a uniform random variable is an interval with finite limits, and any two intervals of equal size within the sample space have equal probability. The following ex ample, which is used throughout this chapter, examines this random variable by defining an experiment in which the procedure is to spin a pointer in a circle of circumference one meter. (This model is very similar to the model of the phase angle of the signal that arrives at the radio receiver of a cellular telephone, with the sample space ranging between 0 and 33 2 π radians instead of 0 and 1 meter.) This example will demonstrate that all outcomes have probability zero....
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This document was uploaded on 10/28/2010.
 Spring '09

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