chap05PRN

chap05PRN - Chapter 5 Continuous Random Variables A...

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Unformatted text preview: 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: with )x X(P )x X(P )x(F ? d 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 1 The probability density function ( PDF ) is given by: for all values of x . 0)x(f ! 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 1x0 for 1 )x(f A graph of the probability density function is below. 1 1 a f(x) x Area = P(X < a) = F(a) Econ 325 Chapter 5 2 The important properties of the PDF are: x the total area under the PDF is equal to one. x 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 1 b a 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 3 In general, the uniform distribution over the interval [,] has the PDF: min x max x ?? otherwise for xx 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 3/4 1/2 1/4 f(x) x Again, note that the total area under the PDF is equal to one. Econ 325 Chapter 5 4 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 5 Example: Exercise 5.6, page 198 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 4x0 25.0 )x(f Selected questions and answers: 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 4 3 2 1 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 Econ 325 Chapter 5 6 The rescue teams base is at the mid-point of this stretch of river. Find the probability that a given emergency arises more than...
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chap05PRN - Chapter 5 Continuous Random Variables A...

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