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U7 Continuous Random Variables 1 Module U7 Continuous Random Variables Primary Author: Email Address: Last Update: Prerequisite Competencies: Module Objectives: James D. McCalley, Iowa State University [email protected] 7/12/02 Discrete Random Variables, Module U6 1. To distinguish continuous random variables from discrete random variables. 2. To explain the relationship between a probability density function and a probability mass function. 3. To obtain cumulative distribution functions from probability density functions and vice-versa. 4. To use probability density functions and cumulative distribution functions to obtain probabilities. Overview A continuous random variable (RV), like a discrete RV, is a real numerically valued function of the experimental outcomes that is defined over a sample space. It is the nature of the sample space that determines which type of RV it is; RVs with countable sample spaces are discrete; RVs with non-countable sample spaces are continuous. U7.1 Probability Density Functions In the discrete case, we used the probability mass function (PMF), f X x to characterize the relation between the probability of the RV X taking on the value x. This probability was given as an explicit non-zero numerical value. However, if we were to use a continuous RV, then we allow that our RV X may take on any value x min x x max . Since there are an infinite number of possible values that X may take, the probability that X is exactly equal to a particular value, say 2.43985, is zero. f X x lim x 0 Px X x x x (U7.1) U7 Continuous Random Variables 2 Probability density functions can take a variety of shapes. To illustrate, let’s look at two fairly simple PDFs, the uniform PDF and the triangular PDF. 0.02, f V v 0, 475 V 525 otherwise (U7.2) Figure U7.1 Uniform PDF for Instrument Measurement Error Example. U7 Continuous Random Variables 3 0.16 d 0.8, 5 d 7.5 f D d 0.16 d 1.6, 7.5 d 10 0, otherwise (U7.3) Figure U7.2 Triangular PDF for Load Level Sample One may observe from the above two examples that 1. f X x 0 ( probability per unit RV, must be non-negative) 2. U7.2 f X x 1 (total probability, must be 1) Evaluation of Probabilities Using PDFs b Pa X b f X x dx a (U7.7) U7 Continuous Random Variables 4 Recall the uniform distribution of Fig U7.1. 1. Between 475 and 525: P475 V 525 525 525 475 f V v dv 475 0.02 dv 1.0 2. Between 475 and 505:1: P475 V 505 505 505 f V v dv 475 0.02 dv 0.6 475 3. Between 450 and 505: P450 V 505 505 505 f vdv 0.02dv 0.6 P475 V 505 V 450 475 4. Less than 505: P V 505 505 505 f V v dv 475 0.02 dv 0.6 5. Greater than 505: P505 V 525 505 f V v dv 505 0.02 dv 0.4 All of the above calculations may be interpreted as areas under the curve of Figure (U7.1) U7.3 Cumulative Distribution Functions P X b P X b This probability has been given a specific name, the cumulative probability function (CDF), denoted by F X x and given by FX x P X x x f X d (U7.9) It is analogous to the CDF defined for the discrete case in that it gives P X x . U7 Continuous Random Variables Example IM4: The uniform pdf case. 0, v 475 FV v 0.02 v 9.5, 475 v 525 1.0, 525 v Figure U7.3 CDF for Instrument Measurement Error Example 5 U7 Continuous Random Variables 6 Example LL4: Triangular pdf case. 0, 2 0.08 d 0.8d 2, FD d 0.5 0.08 d 2 1.6d 7.5, 1.0, d 5 5 d 7. 5 7.5 d 10 10 d Figure U7.4 CDF for Load Level Example Important properties of all continuous CDFs. 1. Its lower bound is zero, FX 0 , which must be the case since FX P X 0 2. Its upper bound is one, FX 1 , which must be the case since FX P X 1 3. All CDFs must be non-decreasing functions. P X x P X x 1 P X x 1 FX x (U7.10) 4. 5. 6. Pa X b FX b FX a d FX x f X x dx