{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter 4 2114 - Continuous Random Variables and...

Info icon This preview shows pages 1–17. Sign up to view the full content.

View Full Document Right Arrow Icon
Continuous Random Variables and Probability Distributions Chapter 4
Image of page 1

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

View Full Document Right Arrow Icon
2 Continuous Random Variables Continuous random variables can assume infinitely many values corresponding to points in a particular interval. A random variable X is continuous if its set of possible values is an entire interval of numbers (If A < B , then any number x between A and B is possible). If the measurement scale of X can be subdivided to any extent desired, then a variable is continuous, if it cannot, the variable is discrete.
Image of page 2
Examples of Continuous Random Variable If measurement scale can be subdivided to any extent possible, variable is continuous X = Depth measurement of a lake (A = minimum depth, B= max depth) X = pH of a chemical compound (range 0- 14, but any value in between is possible) Discrete measurements often well-modeled by continuous functions
Image of page 3

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

View Full Document Right Arrow Icon
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.005 0.01 0.015 0.02 0.025 0.03 Say you measured the depth of a lake to the nearest meter and then measured it to the nearest cm and built probability histograms: limit A B Meters Cm
Image of page 4
Limit of a sequence of discrete histograms (probability density function) 0 0.005 0.01 0.015 0.02 0.025 0.03 A B f(x)= 9-x 2
Image of page 5

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

View Full Document Right Arrow Icon
6 Probability Distribution Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is a function f ( x ) such that for any two numbers a and b , ( 29 ( ) b a P a X b f x dx = The graph of f is the density curve .
Image of page 6
7 Probability Density Function For f ( x ) to be a pdf 1. f ( x ) > 0 for all values of x . 2.The area of the region between the graph of f and the x – axis is equal to 1. Area = 1 ( ) y f x =
Image of page 7

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

View Full Document Right Arrow Icon
8 Probability Density Function is given by the area of the shaded region. ( ) y f x = b a ( ) P a X b
Image of page 8
9 Uniform Distribution A continuous rv X is said to have a uniform distribution on the interval [ A , B ] if the pdf of X is ( 29 1 ; , 0 otherwise A x B f x A B B A = -
Image of page 9

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

View Full Document Right Arrow Icon
A B - 1 f(x) A B pdf of Uniform Distribution Area = 1 x Example…
Image of page 10
A read/write head for a hard disk has to locate a record as the disk rotates once every 25 ms. Let X = time it takes to locate the record. Assume that X is uniformly distributed on the interval [0,25]. What is: P(10 < X < 20)? P(X > 10)? 11
Image of page 11

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

View Full Document Right Arrow Icon
For a Non-uniform Distribution: Suppose the error involved in making a measurement is a continuous rv X with pdf: f(x)= .09375(4-x 2 ) -2 < x < 2 0 otherwise Compute P(X> 0). Compute P(-1< X< 1). Compute P(-.5< X`OR X > .5). 12
Image of page 12
13 Probability for a Continuous rv If X is a continuous rv, then for any number c , P ( x = c ) = 0. For any two numbers a and b with a < b, ( ) ( ) P a X b P a X b = < ( ) P a X b = < ( ) P a X b = < < Example…
Image of page 13

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

View Full Document Right Arrow Icon
14 The Cumulative Distribution Function The cumulative distribution function, F ( x ) for a continuous rv X is defined for every number x by ( 29 ( ) ( ) x F x P X x f y dy -∞ = = For each x , F ( x ) is the area under the density curve to the left of x .
Image of page 14
15 Using F ( x ) to Compute Probabilities ( 29 ( ) ( ) P a X b F b F a = - Let X be a continuous rv with pdf f ( x ) and cdf F ( x ). Then for any number a , and for any numbers a and b with a < b, ( 29 1 ( ) P X a F a = - Thus if the pdf can be integrated, we can easily find probabilities for continuous rvs
Image of page 15

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

View Full Document Right Arrow Icon
CDF for the Uniform Distribution dy A B dy y f x F x x A - - = = 1 ) ( ) ( A B A x A B A A B x y A B x A - - = - - - = - = 1 16
Image of page 16
Image of page 17
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern