chap05PRN

# chap05PRN - Chapter 5 Continuous Random Variables A...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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) ȉ (width). The answer is: 0.25 0) (0.25)(1 )1(F)1 X(P Econ 325 – Chapter 5 6 ¡ 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...
View Full Document

## This note was uploaded on 01/28/2011 for the course ECON 325 taught by Professor Whistler during the Spring '10 term at UBC.

### Page1 / 18

chap05PRN - Chapter 5 Continuous Random Variables A...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online