slide6

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

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

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

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

View Full Document
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
Mann E6 3/48

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

View Full Document
Mann E6 4/48
Mann E6 5/48

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

View Full Document
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
Mann E6 7/48

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

View Full Document
Mann E6 8/48
Mann E6 9/48

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

View Full Document
Mann E6 10/48
Mann E6 11/48

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

View Full Document
Mann E6 12/48
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

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

View Full Document
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
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

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.

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern