slide6 - Chapter 6 Continuous Random Variables and the...

Info icon This preview shows pages 1–16. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 6: Continuous Random Variables and the Normal Distributions Bin Wang [email protected] Department of Mathematics and Statistics University of South Alabama Mann E6 1/48
Image of page 1

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

View Full Document Right Arrow Icon
6.1 Continuous Probability Distribution A continuous random variable can assume any value over an interval or intervals. p.m.f won’t work for continous random varibales. Mann E6 2/48
Image of page 2
Mann E6 3/48
Image of page 3

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

View Full Document Right Arrow Icon
Mann E6 4/48
Image of page 4
Mann E6 5/48
Image of page 5

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

View Full Document Right Arrow Icon
Probability density function The probability distribution of a continuous random variable possesses the following two characteristics: 1 The probability that x assumes a value in any interval lies in the range 0 to 1; 2 The total probability of all the (mutually exclusive) intervals within which x can assume a value is 1.0. Mann E6 6/48
Image of page 6
Mann E6 7/48
Image of page 7

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

View Full Document Right Arrow Icon
Mann E6 8/48
Image of page 8
Mann E6 9/48
Image of page 9

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

View Full Document Right Arrow Icon
Mann E6 10/48
Image of page 10
Mann E6 11/48
Image of page 11

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

View Full Document Right Arrow Icon
Mann E6 12/48
Image of page 12
Mean and Variance Definition (Mean and Variance) Suppose X is a continuous random variable with probability density function f ( x ). The mean or expected value of X , denoted as μ or E ( X ), is μ = E ( X ) = Z -∞ xf ( x ) dx The variance of X , denoted as V ( X ) or σ 2 , is σ 2 = V ( X ) = Z -∞ ( x - μ ) 2 f ( x ) dx = Z -∞ x 2 f ( x ) dx - μ 2 The standard deviation of X is σ = [ V ( X )] 1 / 2 . Mann E6 13/48
Image of page 13

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

View Full Document Right Arrow Icon
6.2 The Normal Distribution Definition (Normal Distribution) A random variable X with probability density function f ( x ) = 1 2 πσ e - ( x - μ ) 2 2 σ 2 for - ∞ < x < has a normal distribution ( and is called a normal random variable ) with parameters μ and σ , where -∞ < μ < , and σ > 0. Also, E ( X ) = μ and V ( X ) = σ 2 Mann E6 14/48
Image of page 14
6.2 The Normal Distribution Definition (Normal Distribution) A random variable X with probability density function f ( x ) = 1 2 πσ e - ( x - μ ) 2 2 σ 2 for - ∞ < x < has a normal distribution ( and is called a normal random variable
Image of page 15

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

View Full Document Right Arrow Icon
Image of page 16
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