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Distribution Basics

Statistics

• Descriptive statistics – Those methods involving the collection, summarisation, presentation

and characterisation of a set of data in order to properly describe the various features of that

set of data.

• Inferential statistics – Those methods that make possible the estimation of a characteristic of

a population or the making of a decision concerning a population based only on a sample.

Population vs. sample

• A population is a total collection of observations or measurements of interest

• A sample is a subset of measurements or observations from a population

Sample vs. population

m = Population mean

s = Population standard deviation

EstimateSamplestatistics

Populationparameters

SAMPLE POPULATION

s = Sample standard deviation (s)̂

X = Sample mean (m)̂

Describing the sample

• Central tendency – a distinct tendency to cluster about a central point (an average)

• Dispersion – amount of variation or spread in the data

• Shape – manner in which data are distributed

Statistical functions

• f(x) – Probability Density Function (PDF)

• Models a histogram

• F(x) – Cumulative Distribution Function (CDF)

• The area under f(x) to the left of x

• Probability of less than: integral of f(x) from negative infinity to x

• R(x) – Reliability function

• The area under f(x) to the right of x

• 1-F(x)

• Probability of greater than: integral of f(x) from x to infinity

• h(x) – Hazard function

• Instantaneous failure rate: f(x)/R(x)

• A measure of proneness to fail

Probability Density Function Definition

Symmetric vs. skewed data

• Mean > Median: Positive or right-skewness

• Mean = Median: Symmetry or zero-skewness

• Mean < Median: Negative or left-skewness

positively skewed (right) negatively skewed (left)

symmetric

Moment Generating Functions

1st moment generating function about the origin

2nd moment generating function about the mean

Skewness = the 3rd moment generating function about the origin

Example• Given the density function f(x) = ax, 0 < x < 10

• Determine the value of a that makes f(x) a valid density function

• Determine the mean and variance of the distribution

• Derive expressions for the reliability and hazard functions

( ) ( )( )ax dxa

xa

a0

102

0

10

2 21

21

210 0 1

1

50 =  =  − =  =

( )E x xx

dxx

dx( ) .= = = = 50 50

1

15010 6 667

0

102

3

0

10

( ) ( )V x x dx( ) . . .= − = − =1

506 667

1

20010 44 449 5 55

3 2 4

0

10

Example Continued

Probability Density Function (PDF)

0.59

0.71

0.83

0.95

1.07

1.19

1.31

1.43

1.55

1.67

1.79

1.91

2.03

2.15

2.27

2.39

2.51

2.63

0

10

20

30

40

Count

Probability Density Function

0

x

f(x)

a b

6 8Diameter

4 6 80

0.2

0.4

0.6

0.8

1

Diameter

Cu

mu

lati

ve D

istr

ibu

tio

n

40

0.002

0.004

0.006

0.008

Pro

ba

bil

ity

De

nsi

ty

Cumulative Distribution Function (CDF)

Reliability Function

0x

1

F(x) R(x)

Hazard Function

• Instantaneous failure rate

## How it Works

1. Clіck оn the “Place оrder tab at the tоp menu оr “Order Nоw” іcоn at the bоttоm, and a new page wіll appear wіth an оrder fоrm tо be fіlled.
2. Fіll іn yоur paper’s іnfоrmatіоn and clіck “PRІCE CALCULATІОN” at the bоttоm tо calculate yоur оrder prіce.
3. Fіll іn yоur paper’s academіc level, deadlіne and the requіred number оf pages frоm the drоp-dоwn menus.
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6. Frоm there, the payment sectіоns wіll shоw, fоllоw the guіded payment prоcess, and yоur оrder wіll be avaіlable fоr оur wrіtіng team tо wоrk оn іt.

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Meanwhіle, іn case yоu need tо uplоad an attachment accоmpanyіng yоur оrder, clіck оn the “Pendіng” buttоn at the left sіdebar menu оf yоur page, then clіck оn the “Vіew” buttоn agaіnst yоur Order ID and clіck “Fіles” and then the “add fіle” оptіоn tо uplоad the fіle.

Basіcally, іf lоst when navіgatіng thrоugh the sіte, оnce lоgged іn, just clіck оn the “Pendіng” buttоn then fоllоw the abоve guіdelіnes. оtherwіse, cоntact suppоrt thrоugh оur chat at the bоttоm rіght cоrner

NB

Payment Prоcess

By clіckіng ‘PRОCEED TО CHECKОUT’ yоu wіll be lоgged іn tо yоur accоunt autоmatіcally where yоu can vіew yоur оrder detaіls. At the bоttоm оf yоur оrder detaіls, yоu wіll see the ‘Checkоut” buttоn and a checkоut іmage that hіghlіght pоssіble mоdes оf payment. Clіck the checkоut buttоn, and іt wіll redіrect yоu tо a PayPal page frоm where yоu can chооse yоur payment оptіоn frоm the fоllоwіng;

1. Pay wіth my PayPal accоunt‘– select thіs оptіоn іf yоu have a PayPal accоunt.
2. Pay wіth a debіt оr credіt card’ or ‘Guest Checkout’ – select thіs оptіоn tо pay usіng yоur debіt оr credіt card іf yоu dоn’t have a PayPal accоunt.
3. Dо nоt fоrget tо make payment sо that the оrder can be vіsіble tо оur experts/tutоrs/wrіters.

Regards,

Custоmer Suppоrt
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