Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 Slide 1

Continuous Probability Distributions

• Many continuous probability distributions, including: UniformNormalGammaExponentialChi-SquaredLognormalWeibull

Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 Slide 2

Normal Distribution

• The “bell-shaped curve”• Also called the Gaussian distribution• The most widely used distribution in statistical

analysis forms the basis for most of the parametric tests we’ll

perform later in this course.describes or approximates most phenomena in

nature, industry, or research

• Random variables (X) following this distribution are called normal random variables. the parameters of the normal distribution are μ and σ

(sometimes μ and σ2.)

Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 Slide 3

Normal Distribution• The density function of the normal random

variable X, with mean μ and variance σ2, is

all x.2

2

2

)(

2

1),;(

x

exn

Normal Distribution

00.05

0.10.150.2

0.25

0.30.35

0 2 4 6 8 10 12

x

P(x

)

(μ = 5, σ = 1.5)

Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 Slide 4

Standard Normal RV …

• Note: the probability of X taking on any value between x1 and x2 is given by:

• To ease calculations, we define a normal random variable

where Z is normally distributed with μ = 0 and σ2 = 1

dxedxxnxXxPx

x

xx

x

2

1

2

22

1

2

)(

212

1),;()(

X

Z

Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 Slide 5

Standard Normal Distribution

Standard Normal Distribution

-5 -4 -3 -2 -1 0 1 2 3 4 5

Z

• Table A.1 : “Areas under the standard normal curve from -

∞ to z”Page 915 negative values for zPage 916 positive values for z

Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 Slide 6

Examples

• P(Z ≤ 1) =

• P(Z ≥ -1) =

• P(-0.45 ≤ Z ≤ 0.36) =

-5 0 5

-5 0 5

-5 0 5

Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 Slide 7

Your turn …

• Use Table A.1 to determine (draw the picture!)

1. P(Z ≤ 0.8) =

2. P(Z ≥ 1.96) =

3. P(-0.25 ≤ Z ≤ 0.15) =

4. P(Z ≤ -2.0 or Z ≥ 2.0) =

Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 Slide 8

Applications of the Normal Distribution• A certain machine makes electrical resistors having a mean

resistance of 40 ohms and a standard deviation of 2 ohms. What percentage of the resistors will have a resistance less than 44 ohms?

• Solution: X is normally distributed with μ = 40 and σ = 2 and x = 44

P(X<44) = P(Z< +2.0) = 0.9772 Therefore, we conclude that 97.72% will have a resistance less

than 44 ohms. What percentage will have a resistance greater than 44 ohms?

-5 0 5

2

4044

Z

XZ

Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 Slide 9

Terminology Used in ISE 327 Text• A certain machine makes electrical resistors having a

mean resistance of 40 ohms and a standard deviation of 2 ohms. What percentage of the resistors will have a resistance greater than 44 ohms?

• Solution: X is normally distributed with μ x= 40 and σx = 2 and x = 44

P(X>44) = 1 - P(Z< +2.0) = 1 - 0.9772 Therefore, we conclude that 2.28% will have a resistance

greater than 44 ohms.

-5 0 5

2

4044

Z

XZ

x

x

Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 Slide 10

Your Turn

1. What is the probability that a single resistor will have a rating between 42 and 44 ohms?

2. Specifications are that the resistors are 40 ± 3 ohms. What percentage of the resistors will be within specifications?

-5 0 5

DRAW THE PICTURE!!

-5 0 5

Statistical Review for Chapters 3 and 4 ISE 327 Fall 2008 Slide 11

The Normal Distribution “In Reverse”• Example:

Given a normal distribution with μ = 40 and σ = 6, find the value of X for which 45% of the area under the normal curve is to the left of X.

1) If P(Z < k) = 0.45,

k = ___________

2) Z = _______

X = _________

-5 0 5