1 ch5. probability densities dr. deshi ye yedeshi@zju.edu.cn

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Ch5. Probability Densities

Dr. Deshi Ye

yedeshi@zju.edu.cn

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Outline Continuous Random variables Kinds of Probability distribution

Normal distr. Uniform distr. Log-Normal dist. Gamma distr. Beta distr. Weibull distr.

Joint distribution Checking data if it is normal?

Transform observation to near normal Simulation

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5.1 Continuous Random Variables

Continuous sample space: the speed of car, the amount of alcohol in a person’s blood

Consider the probability that if an accident occurs on a freeway whose length is 200 miles.

Question: how to assign probabilities?

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Assign Prob.

Suppose we are interested in the prob. that a given random variable will take on a value on the interval [a, b]

We divide [a, b] into n equal subintervals of width ∆x, b – a = n ∆x, containing the points x1, x2, ..., xn, respectively.

Then

n

ii xxfbxaP

1

)()(

Frequency

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If f is an integrable function for all values of the random variable, letting ∆x-> 0, then

b

adxxfbxaP )()(

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Continuous Probability Density Function

1. Shows All Values, x, & Frequencies, f(x) f(X) Is Not Probability

2. Properties

(Area Under Curve)(Area Under Curve)

ValueValue

(Value, Frequency)(Value, Frequency)

FrequencyFrequency

f(x)f(x)

aa bbxx

ff xx dxdx

ff xx

(( ))

(( ))

All All XX

aa x x bb

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0,0,

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Continuous Random Variable Probability

Probability Is Area Probability Is Area Under Curve!Under Curve!

f(x)f(x)

Xa b

b

adxxfbxaP )()(

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Distribution function F

Distribution function F (cumulative distribution )

xdttfxF )()(

)( xXP Or

( )( )

dF xf x

x

Integral calculus:

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EX

If a random variable has the probability density

find the probabilities that it will take on a value

A) between 1 and 3 B) greater than 0.5

else

xforexf

x

0

02)(

2

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Solution

133.0|2)31( 2631

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1

2 eeedxexP xx

2 2 10.50.5

( 0.5) 2 | 0 0.368x xP x e dx e e

B)

A)

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Mean and Variance

( )xf x dx

Mean:

Variance:

2 2( ) ( )x f x dx

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K-th moment

About the original

About the mean

dxxfxkk

)('

( ) ( )kk x f x dx

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Useful cheat

dxexa

n

a

exdxex axn

axnaxn 1

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Continuous Probability Distribution Models

Continuous Probability Distribution

Uniform Normal Exponential Others

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Normal Distribution

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5.2 The Normal Distribution

Normal probability density (normal distribution)

xexf

x2

2

2

)(2

2

1),;(

The mean and variance of normal distribution is exactly

2 and

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The Normal Distribution

X

f(X)

X

f(X)

Mean Mean Median Median ModeMode

1. ‘Bell-Shaped’ & Symmetrical

2. Mean, median, mode are equal

3. Random variable has infinite range

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The Normal Distribution

f(x) = Frequency of random variable x = Population standard deviation = 3.14159; e = 2.71828x = value of random variable (- < x < ) = Population mean

xexf

x2

2

2

)(2

2

1),;(

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Effect of varying parameters ( & )

X

f(X)

CA

B

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Standard normal distribution function

Standard normal distribution, with mean 0 and variance 1. Hence

z t dtezFzZP 2/2

2

1)()(

)()()( aFbFbxaP

( ) 1 ( )F z F z

Normal table

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Standardize theNormal Distribution

X

X

One table! One table!

Normal DistributionNormal Distribution

= 0

= 1

Z = 0

= 1

Z

ZX

ZX

Standardized

Normal DistributionStandardized

Normal Distribution

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Not standard normal distribution

Let , then the random

Variable Z, F(z) has a standard normal distribution. We call it z-scores.

When X has normal distribution with mean and standard deviation

uX

Z

)()()(

aF

bFbxaP

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Find z values for the known probability

Given probability relating to standard normal distribution, find the corresponding value z.

F(z) is known, what is the value of z?

Let be such that probability is

where

z

)( zZP

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Finding Z Values for Known Probabilities

Z .00 0.2

0.0 .0000 .0040 .0080

0.1 .0398 .0438 .0478

0.2 .0793 .0832 .0871

.1179 .1255

Z .00 0.2

0.0 .0000 .0040 .0080

0.1 .0398 .0438 .0478

0.2 .0793 .0832 .0871

.1179 .1255

Z = 0

= 1

.31 Z = 0

= 1

.31

.1217.1217.1217.1217.01.01

0.30.3 .1217

Standardized Normal Standardized Normal Probability Table (Portion)Probability Table (Portion)

Standardized Normal Standardized Normal Probability Table (Portion)Probability Table (Portion)

What is Z given What is Z given P(Z) = .1217?P(Z) = .1217?What is Z given What is Z given P(Z) = .1217?P(Z) = .1217?

Shaded area Shaded area exaggeratedexaggeratedShaded area Shaded area exaggeratedexaggerated

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1)(zF

Find the following values (check it in Table)

645.1,95.005.01)(

33.2,99.001.01)(

05.005.0

01.001.0

zzF

zzF

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5.3 The Normal Approximation to the binomial distribution

Theorem 5.1. If X is a random variable having the binomial distribution with parameter n and p, the limiting form of the distribution function of the standardized random variable

as n approaches infinity, is given by the standard normal distribution

(1 )

X npZ

np p

zdtezFz t 2/2

2

1)(

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EX If 20% of the memory chips made in a

certain plant are defective, what are the probabilities that in a lot of 100 random chosen for inspection?

A) at most 15.5 will be defective B) exactly 15 will be defective

Hint: calculate it in binomial dist. And normal distribution.

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A good rule

A good rule for normal approximation to the binomial distribution is that both

np and n(1-p) is at least 15