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chap05PRN econ 325

chap05PRN econ 325 - Chapter 5 Continuous Random Variables...

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Econ 325 – Chapter 5 1 Chapter 5 Continuous Random Variables A continuous random variable can take any numerical value in some interval. Assigning probabilities to individual values is not possible. Probabilities can be measured in a given range. For a continuous random variable X with a numerical value of interest x the cumulative distribution function ( CDF ) is denoted by: ) x X ( P ) x X ( P ) x ( F < = = with 0 ) x X ( P = = For two numerical values a and b , with a < b , the probability that the outcome is in a range is: ) a ( F ) b ( F ) a X ( P ) b X ( P ) b X a ( P ) b X a ( P - = < - < = = < < Econ 325 – Chapter 5 2 The probability density function ( PDF ) is given by: 0 ) x ( f for all values of x . The properties of a probability density function can be illustrated with a special distribution called the uniform distribution . The uniform distribution over the interval [0, 1] has the PDF: < < = otherwise 1 x 0 for 0 1 ) x ( f A graph of the probability density function is below. 1 0 1 a 0 f(x) x Area = P(X < a) = F(a)
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Econ 325 – Chapter 5 3 The important properties of the PDF are: the total area under the PDF is equal to one. the area under the PDF to the left of the value a is ) a ( F . The next graph illustrates that the PDF can also be used to find a range probability. 1 0 1 b a 0 f(x) x Area = P(a < X < b) = F(b) - F(a) The range probability ) b X a ( P < < is the area under the PDF between the values a and b . Econ 325 – Chapter 5 4 In general, the uniform distribution over the interval [ min x , max x ] has the PDF: < < - = otherwise for 0 x x x x x 1 ) x ( f max min min max For example, consider the uniform distribution over the interval ] , [ 4 3 4 1 . A graph of the probability density function is below: 2 1 0 3/4 1/2 1/4 0 f(x) x Again, note that the total area under the PDF is equal to one.
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Econ 325 – Chapter 5 5 By comparing the graphs of the PDFs for the uniform distribution over the interval [0, 1] and the uniform distribution over ] , [ 4 3 4 1 it can be seen that both are centered at 2 1 . However, the two distributions have different dispersion. That is, the PDF for the uniform distribution over ] , [ 4 3 4 1 has a higher peak to suggest smaller dispersion. Econ 325 – Chapter 5 6 Example An emergency rescue team operates on a 4-mile stretch of river. Let the random variable X be the distance (in miles) of an emergency from the northernmost point of this stretch of river. X follows a uniform distribution over the interval [0, 4] with PDF: < < = otherwise for 0 4 x 0 25 . 0 ) x ( f Questions and answers: square4 Find the probability that a given emergency arises within one mile of the northernmost point of this stretch of river. A graph of the PDF illustrates the problem: 0.25 0 4 3 2 1 0 f(x) x Area = P(X < 1) = F(1) The area of a box is calculated as: (height)∙(width). The answer is: 0.25 0) (0.25)(1 = - = = < ) 1 ( F ) 1 X ( P
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Econ 325 – Chapter 5 7 square4 The rescue team’s base is at the mid-point of this stretch of river. Find the probability that a given emergency arises more than 1.5 miles from this base.
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