{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture7_Basic Probability

# Lecture7_Basic Probability - Lecture 8 Basic Probability...

This preview shows pages 1–13. Sign up to view the full content.

Lecture 8 Basic Probability Concepts

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

View Full Document
Normal Distribution ( 29 2 2 2 1 P( ) 2 where is the mean value of the variable and is the standard deviation of the variable Probability density function for a normal distribution is give . y n b y y y y y y y e μ σ σ π μ σ - - = P( ) y y y σ - y σ μ
Web Based Probability Calculator

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

View Full Document
Transformation From The Normal Distribution To The Standard Normal Distribution Assume we have a of the variable with: mean value of standard de Normal Distribution Standard Normal Distr viation of The is the transformation from to via: ibution Th e y y y y y y y y z y z μ σ μ σ = = - = Standard Normal Distribution has a mean of zero and a standard deviation of 1!
* * What is the probability , where is some specific value of ? y y y y * * Pr( ) Pr( ) y y z z =

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

View Full Document
Z Table (Posted on WebCT)
Z Table Example - z* = 0.75

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

View Full Document
Some Z Table Values (Table 5.1 of Hyman) z* One sided probability Two sided probability 0 0.5 1.0 0.5 0.308 0.616 1.0 0.159 0.318 1.5 0.067 0.134 2.0 0.023 0.046 2.5 0.0062 0.0124 3.0 0.00135 0.0027 1 σ 2 σ 3 σ
USL, LSL, and Tolerance LSL = Lower specification limit, USL = Upper specification limit Standard tolerance values assume: Tolerance 3 or 3 y y y y σ σ = ± ∆ = ± = μ LSL -3 σ USL 3 σ

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

View Full Document
Resistor Example ( 29 4 A batch of resistors is specified as 10 Kohm 5% Find the tolerance R, and . R 10 0.05 500 ohms 10 Kohm Assume the tolerance, R, represents 3 , which implies that for a large batch of resistors, al μ σ μ σ ± = = = l but 0.3% will fall between 9.5 Kohm and 10.5 Kohm, and R 500 167 ohms 3 3 σ = = =
Central Limit Theorem 1 2 3 i i 1 2 3 Let , , ... be a set if random variables for which both the expected (mean) values, , and standard deviations, , exist and are finite. Assume is a function of , , ... . The c n n x x x x f x x x x μ σ entral limit theorem states that the probability distribution of will converge to a normal distribution as becomes large. f n

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

View Full Document
History of Central Limit Theorem The central limit theorem has an interesting history. The first version of this theorem was postulated by the French-born English mathematician Abraham de Moivre, who, in a remarkable article published in 1733 , used the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This finding was far ahead of its time, and was nearly forgotten until the famous French mathematician Pierre-Simon Laplace rescued it from obscurity in his monumental work Théorie Analytique des Probabilités, which was published in 1812. Laplace expanded De Moivre's finding by approximating the binomial distribution with the normal distribution. But as with De Moivre, Laplace's finding received little attention in his own time. It was not until the nineteenth century was at an end that the importance of the central limit theorem was discerned, when, in 1901 , Russian mathematician Aleksandr Lyapunov defined it in general terms and proved precisely how it worked mathematically. Nowadays, the central limit
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