[PDF] ECE 302: Lecture 43 Cumulative Distribution Function





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f(x)= 2x ? 3x +5x ?1 f (x)= 3×2x ?2× 3x +5

2. +5x ?1 f '(x)= 3×2x. 2. ?2× 3x +5. Définition : Soit f une fonction polynôme du troisième degré définie sur ? par f(x) = ax3 +bx2 + cx + d .



f(x)= 5x ? 3x +2 f (x)= 2×5x ? 3

3x +2 f '(x)= 2×5x ? 3. Définition : Soit f une fonction polynôme du second degré définie sur ? par f(x) = ax2 +bx + c . On appelle fonction dérivée de f 



SECOND DEGRE (Partie 2)

L'équation f(x)=0 a deux solutions donc la courbe de f traverse l'axe des abscisses en deux points. Page 4. 4. Yvan Monka – Académie de Strasbourg – www.maths- 



FONCTION DERIVÉE

Ainsi pour tout x de R {0}



FONCTIONS AFFINES (Partie 2)

2. 9. – 1 donc C ? (d). Soit une fonction affine f : x ax + b représentée dans un repère par une droite d. Les coordonnées (x ; y) d'un point M appartenant 



SECOND DEGRÉ (Partie 1)

On veut exprimer la fonction f sous sa forme canonique : f (x) = ?(x - ?)2 + ? où ? ? et ? sont des nombres réels. f (x) = 2x2 ? 20x +10. = 2 x2 ?10x.



FONCTION EXPONENTIELLE

f est donc croissante sur l'intervalle et décroissante sur l'intervalle . On dresse le tableau de variations : x. 2. +. 0. -. 0 x 



NOMBRE DERIVÉ

f (x) = L et on lit : "La limite de f (x) lorsque x tend vers 0 est égale à L. II. Dérivabilité. 1) Rappel : Coefficient directeur d'une droite.





FONCTIONS COSINUS ET SINUS

2) sin(?x) = ?sinx. Remarque : On dit que la fonction cosinus est paire et que la fonction sinus est impaire. Définitions : Une fonction f est paire 



FX Series Programmable Controllers Foreword ENG

Algebra 2-Trig Name_____ Unit 1: Lesson 3 Transformations of Graphs Hour_____ Graph the following functions without using technology Feel free to use a graphing calculator to check your answer but you should be able to look at the function and apply what you learned



ECE 302: Lecture 43 Cumulative Distribution Function

fX(x) = Therefore the overall PDF is 0 fX(x) =34 12e?2x 0 3= 4 =e?2x Summary Thecumulative distribution function (CDF)of Xis FX(x)def=P[X?x] CDF must satisfy theseproperties:Non-decreasing FX(??) = 0 andFX(?) = 1 P[a?X?b] =FX(b)?FX(a) Right continuous: Solid dot on at the start If discontinuous at b thenP[X=b] = Gap

What are the FX and fx2cont?

Les appareils FX et FX2Cont été conçus de manière à assurer un câblage simple et sûr. Si lors de leur installation des incertitudes persistent, n’hésitez pas à consulter un électricien compétent qualifié et formé à l’utilisation des normes électrotechniques locales et nationales. FRE Elektrischer Anschluß

What is the eqn for fx2c?

Eqn 1 for FX: Eqn1 for FX2C: Rb ? 4Rp 15 ? Rp k ? Rb ? 3Rp 13 ? Rp k ? Eqn 2 for FX: Eqn2 for FX2C: Rb ? 6 I ? 1.5 k ? Rb ? 4 I ? 1.0 k ? 5 - 7 FX Series Programmable Controllers Inputs 5.

How does FX2 read and write data?

The FX2 reads host data from an OUT endpoint buffer, and writes data for transmis- sion to the host to an IN endpoint buffer. FX2 contains three 64-byte endpoint buffers, plus 4 Kilobytes of buffer space that can be config- ured various ways, as indicated by Figure 1-16.

Is the FX2 bi-directional?

bi-directional, so the FX2 provides a single 64-byte buffer, EP0BUF, which firmware handles exactly like a bulk endpoint buffer for the data stages of a CONTROL transfer. A second 8-byte buffer called SETUPDAT, which is unique to endpoint zero, holds data that arrives in the SETUP stage of a CONTROL transfer.

©Stanley Chan 2022. All Rights Reserved.ECE 302: Lecture 4.3 Cumulative Distribution Function

Prof Stanley Chan

School of Electrical and Computer Engineering

Purdue University1/21

©Stanley Chan 2022. All Rights Reserved.Outline

Cumulative distribution function(CDF):

F

What are the properties of CDF?

How are CDFs related to PDF?

2/21 ©Stanley Chan 2022. All Rights Reserved.Deifinition

Deifinition

LetXbe a continuous random variable with a sample space Ω =R. The cumulative distribution function (CDF)ofXis F ©Stanley Chan 2022. All Rights Reserved.Example Question. (Uniform random variable) LetXbe a continuous random the CDF ofX.

Solution.

F

X(x) = =

1,x>b.4/21

©Stanley Chan 2022. All Rights Reserved.Example 2 Question.(Exponential random variable) LetXbe a continuous random variable with PDFfX(x) =λe-λxforx≥0, and is 0 otherwise. Find the

CDF ofX.

Solution.

F

X(x) = =(

0,x<0,

1-e-λx,x≥0.5/21

©Stanley Chan 2022. All Rights Reserved.Properties 1-3

Theorem

LetXbe a random variable (either continuous or discrete), then the CDF ofXhas the following properties: (i)The CDF is anon-decreasing. (ii)Themaximumof the CDF is whenx=∞:FX(+∞) = 1. (iii)Theminimumof the CDF is whenx=-∞:FX(-∞) = 0.6/21 ©Stanley Chan 2022. All Rights Reserved.Property 4

Theorem

LetXbe a continuous random variable. If the CDFFXis continuous at ©Stanley Chan 2022. All Rights Reserved.Example Example 1. (Exponential random variable.)fX(x) =λe-λxforx≥0, F (a) PDF approach: (b) CDF approach: 8/21 ©Stanley Chan 2022. All Rights Reserved.Example Example 2. LetXbe a random variable with PDFfX(x) = 2xfor (a) Find CDF. F P 13 = =536 9/21 ©Stanley Chan 2022. All Rights Reserved.Left and Right Continuous

Deifinition

A functionFX(x) is said to beLeft-continuousatx=bifFX(b) =FX(b-)def= limh→0FX(b-h);Right-continuousatx=bifFX(b) =FX(b+)def= limh→0FX(b+h);Continuousatx=bif it is both right-continuous and

left-continuous atx=b. In this case, we have lim h→0FX(b-h) = limh→0FX(b+h) =F(b).10/21 ©Stanley Chan 2022. All Rights Reserved.Left and Right Continuous

Figure

The deifinition of left and right continuous at a p ointb. 11/21 ©Stanley Chan 2022. All Rights Reserved.Property 5: CDF must be right continuous

Theorem

For any random variableX(discrete or continuous),FX(x)is always right-continuous. That is, F X(b) =FX(b+)def= limh→0FX(b+h) (4)Figure:A CDF must b erigh tcontinuous 12/21 ©Stanley Chan 2022. All Rights Reserved.Property 6: Jump

Theorem

For any random variableX(discrete or continuous),P[X=b]is

P[X=b] =(

F

X(b)-FX(b-),ifFXis discontinuous atx=b

0,otherwise.(5)13/21

©Stanley Chan 2022. All Rights Reserved.Example

Example. Consider a random variableXwith a PDF

f

X(x) =

12 ,x= 3,

0,otherwise.

Find CDF.

F

X(x) = =x22

F

X(x) = =12

14/21 ©Stanley Chan 2022. All Rights Reserved.Example

Figure

An example of converting a PDF to a CDF.

15/21 ©Stanley Chan 2022. All Rights Reserved.Example (c)x= 3: F

X(3) = = 1,x= 3.

(d)x>3: F

X(x) = = 1,x>3.

Therefore,

F

X(x) =

0,x<0, x 22
12

1,x≥3.

16/21 ©Stanley Chan 2022. All Rights Reserved.Retrieving PDF from CDF

Theorem

Theprobability density function(PDF) is the derivative of the cumulative distribution function (CDF): f

X(x) =dFX(x)dx

=ddx Z x f

X(x′)dx′,(6)

providedFXis diffferentiable atx. IfFXis not diffferentiable atx, then, f

X(x) =P[X=x] =FX(x)-limh→0FX(x-h).(7)17/21

©Stanley Chan 2022. All Rights Reserved.Example

Consider a CDF

F

X(x) =(

0,x<0,

1-14 e-2x,x≥0. Find PDFfX(x).Figure:An example of converting a PDF to a CDF. 18/21 ©Stanley Chan 2022. All Rights Reserved.Example (a) Whenx<0: f

X(x) = = 0

(b) Whenx= 0: f

X(x) = =34

(c) Whenx>0: f

X(x) = =12

e-2x

Therefore, the overall PDF is

f

X(x) =

0,x<0, 34
,x= 0, 12 e-2x,x>0. 19/21 ©Stanley Chan 2022. All Rights Reserved.Summary

Thecumulative distribution function (CDF)ofXis

F

If discontinuous atb, thenP[X=b] = Gap.Relationshipbetween CDF and PDF:PDF→CDF: IntegrationCDF→PDF: Diffferentiation20/21

©Stanley Chan 2022. All Rights Reserved.Questions? 21/21
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