CHAPTER 5 CONTINUOUS PROBABILITY DISTRIBUTION Normal Distributions.

CHAPTER 5CONTINUOUS PROBABILITY DISTRIBUTION

Normal Distributions

The most important of all continuous probability distribution

Used extensively as the basis for many statistical inference methods.

The probability density function is a bell-shaped curve that is symmetric about

The notation X ~ N (2denotes the random variable X has a

normal distribution with mean and variance 2.

A normal distribution with mean and variance 2 = 1 is known as the standard normal distribution.

)1,0(~ NZ

Jan 2009 3

Probability Density function of X ~ N

Graph

.0

,,21)(

221 ]/)[(

andnumberrealaiswhere

xexf x

E(X) = and V(X) = 2

4

In order to compute when X is a normal rvwith parameters and , we must determine

,21 2

21 ]/)[(

x

b

a

e

)( bXaP

However, it is not easy to evaluate this expression, so numerical techniques have been used to evaluate the integral when and , for certain values of a and b and results are tabulated. This table is often used to compute probabilities for any other values of and under consideration.standard normal distribution : The normal distribution with parameter values and .standard normal random variable (Z ) : A random variable having this distribution.

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Probability Density of Standard Normal RV

Cumulative Distribution Function

zezz

,21)( 2

2

zdtedttzzZP

tzz

,21)()()( 2

2

)1,0(~ NZ

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Non standard Normal distribution

When , probabilities involving X are computed using the standard normal table.

)1,0(~ NX

When , probabilities involving X are computed by ‘standardizing’ or ‘linear transformation’. Then

),(~ 2NX

as a standard normal distribution.

XZ

The key idea is that by standardizing, any probability involving X can be expressed as a probability involving a standard normal rv Z, and computed using the standard normal table.

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The function is the area under the standard normal density curve to the left of z

)()( zZPz

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zZPareashaded

Critical Value z: for Z ~ N (0, 1)

denote the value on the z axis for which of the area under the z curve lies to the right of z

CRITICAL VALUES OF THE STANDARD NORMAL Random Variable

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Thus

ab

bXaPbXaP )(

aaXP )(

bbXP 1)(

Important

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Example 0: Evaluate

)25.1() ZPa

)25.1() ZPb )25.138.0() ZPd

8944.03944.05.0)25.1(

1056.0)25.1(1

)25.1(1

ZP )38.0()25.1(

5424.03520.08944.0

)25.1() ZPc

(?)1056.0)25.1(

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micrometermicrometer 05.05.0 a) What is the probability that a line width is greater than 0.62 micrometer?

b) What is the probability that a line width is between 0.47 and 0.62 micrometer?

P(0.47 < X < 0.63) = P(0.6 < Z < 2.6) = P(Z < 2.6) P(Z < 0.6)

= 0.99534 0.27425 = 0.7211

Let X represent the line width

0082.0)4.2(1

)4.2()05.0

5.062.0()62.0(

ZP

ZPZPXP

Example 1:

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b) The line width of 90% of samples is below what value?

Therefore

and x = 0.5641.

9.0)05.0

5.0()(

xZPxXP

28.105.0

5.0

x

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00135.0)3()1.0

4.121.12()1.12(

ZPZPXP

Let X represent the fill volume

.1.0.4.12 ozfluidozfluid

b) If all cans less than 12.1 or greater than 12.6 oz. are scrapped, what proportion of cans is scrapped?

02275.0)2()1.0

4.126.12()6.12(

ZPZPXP

Proportion of cans scrapped = 0.00135 + 0.02275 = 0.0241

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a) At what value should the mean be set so that 99.9% of all cans exceed 12 ounces?

Let X represent the fill volume.1.0 ozfluid

999.0)1.0

12(

999.0)12(

ZP

XP

309.12

09.31.0

12

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Normal Approximation For Binomial Distribution

It is possible to use the normal distribution to approximate binomialprobabilities for cases in which n is large.

If X is a binomial random variable,

)1( pnpnpXXZ

Is approximately a standard normal random variable. Therefore, probabilities computed from Z can be used to approximate probabilities for X.

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)1(

5.0)1(

5.0)(pnpnpm

pnpnpmZPmXP

Normal Approximation For Binomial Distribution

The normal approximation to the binomial distribution is good if

5)1(,5 pnnp

For binomial distribution, E(X) = np and V(X) = np(1- p)

For application purposes, if X is a binomial r.v andand m is a possible value of X

5)1(,5 pnnp

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Example 5 ( Exercise 3.10 pg 112) :

X ~ Bin(n, p), n = 300 (large), p = 0.4. Approximate

0107.0)30.2()6.0()4.0(300

)4.0(3005.0100)100(

ZPXP

)100() XPa

)10080() XPb

0077.0)42.277.4(72

1205.9972

1205.79)10080(

ZP

ZPXP

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X ~ Bin(n, p), n = 300 (large), p = 0.4. Approximate

)130() XPc

1075.0)24.1(172

1205.01301)130(

ZPZPXP