discrete random variables - university of hong kongelec2844/sp21/lec2.pdfx px(0) = 0.2 px(100) =...

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https://www.eee.hku.hk/~elec2844

3+ 4 = 7

x = 7

g(x) = x2 log2

x

P( ) = 1

2

X =

g(X) = X2

log

2

X

P(X = 7) X

P(g(X) = 192)

g(X)

X

X

X

X x

pX

(x) = P(X = x)

X

x

pX

(x) = 1

X =

pX

(x) =

8>>>>>>><

>>>>>>>:

1

4

x = 0

1

2

x = 1

1

4

x = 2

0

pX

(x) X =

pX

(x) p

X = 1

pX

(x) p

X = 1 n

pX

(x) p

X =

pX

(x) X =

[a, b]

pX

(k) =

8><

>:

1

b- a+ 1k = a, a+ 1, . . . , b

0

pX

(k)

ka b

P( ) = p

X =

�1

0

pX

(k) =

�p k = 1

1- p k = 0

pX

(k)

k0 1

n

p

pX

(k) = P(X = k) =

✓n

k

◆pk(1- p)n-k k = 0, 1, . . . , n

pX

(k)

k0 1 2 3 4 5 6

n = 6, p = 0.55

p P( ) = p

pX

(k) = (1- p)k-1p k = 1, 2, . . .

pX

(k)

k1 2 3 4 5 6

p = 0.55

. . .

�

pX

(k) =e-��k

k!k = 0, 1, . . .

� = np n

p

e-��k

k!⇡✓n

k

◆pk(1- p)n-k k ⌧ n

pX

(k)

k0 1 2 3 4 5 6

� = 2

(n, p) � = np

pX

(k) =n!

(n- k)! k!pk(1- p)n-k

=n(n- 1) · · · (n- k+ 1)

nk

· (np)k

k!

✓1-

�

n

◆n-k

k n ! 1 j = 1, . . . , k

n- k+ j

n! 1,

✓1-

�

n

◆-k

! 1,

✓1-

�

n

◆n

! e-�

pX

(k) ! e-��k

k!

X Y = g(X) Y

pY

(y) =X

{x |g(x)=y}

pX

(x)

X

pX

(k) =

8<

:

1

9

k = -4,-3, . . . , 3, 4

0

Y = |X| Z = X2

pY

(0) = pX

(0) = 1

9

pY

(k) = pX

(-k) + pX

(k) = 2

9

k = 1, 2, 3, 4

pY

(k) =

8>>><

>>>:

2

9

k = 1, 2, 3, 4

1

9

k = 0

0

pZ

(0) = pX

(0) = 1

9

pZ

�k2�= p

X

(-k) + pX

(k) = 2

9

k = 1, 2, 3, 4

pZ

(k) =

8>>><

>>>:

2

9

k = 1, 4, 9, 16

1

9

k = 0

0

X

E(X) =X

x

xpX

(x)

X

P(H) = 3

4

X

pX

(k) E(X)

pX

(k) =

8>><

>>:

1

16

k = 03

8

k = 19

16

k = 2

E(X) = 0 · 1

16

+ 1 · 3

8

+ 2 · 9

16

= 3

2

pX

(k)

k0 1 2

n X E(Xn)

= E(X) == E

�X2

�

var(X) = E⇣�

X- E(X)�2

⌘

�X

=p

var(X)

Y = g(X)

E(g(X)) =X

x

g(x)pX

(x)

E(g(X)) = E(Y) =X

y

ypY

(y) =X

y

y

0

@X

{x |g(x)=y}

pX

(x)

1

A

=X

y

X

{x |g(x)=y}

ypX

(x)

=X

y

X

{x |g(x)=y}

g(x)pX

(x)

=X

x

g(x)pX

(x)

E(Xn) =X

x

xnpX

(x)

var(X) =X

x

�x- E(X)

�2

pX

(x)

Y = aX+ b

E(Y) = aE(X) + b

var(Y) = a2

var(X)

E(Y) =X

x

(ax+ b)pX

(x) = aX

x

xpX

(x) + bX

x

pX

(x) = aE(X) + b

var(Y) =X

x

�ax+ b- E(aX+ b)

�2

pX

(x)

=X

x

�ax+ b- aE(X)- b

�2

pX

(x)

=X

x

a2

�x- E(X)

�2

pX

(x) = a2

var(X)

var(X) = E�X2

�-�E(X)

�2

var(X) =X

x

�x- E(X)

�2

pX

(x)

=X

x

�x2 - 2xE(X) + (E(X))2

�pX

(x)

=X

x

x2pX

(x)- 2E(X)X

x

xpX

(x) + (E(X))2X

x

pX

(x)

= E�X2

�- 2�E(X)

�2

+�E(X)

�2

= E�X2

�-�E(X)

�2

P(H) = 3

4

X

pX

(k) E(X)var(X)

Y = (X- E(X))2 = (X- 3

2

)2 x = 0

y = 9

4

x = 1 y = 1

4

x = 2 y = 1

4

pY

(k) =

�1

16

k = 9

4

3

8

+ 9

16

= 15

16

k = 1

4

var(X) = E(Y) = 9

4

· 1

16

+ 1

4

· 15

16

= 3

8

var(X) =2X

x=0

(x- 3

2

)2pX

(x) = 9

4

· 1

16

+ 1

4

· 3

8

+ 1

4

· 9

16

= 3

8

E�X2

�= 02 · 1

16

+ 12 · 3

8

+ 22 · 9

16

= 21

8

var(X) = E�X2

�-�E(X)

�2

= 21

8

- (32

)2 = 3

8

pX

(k)

k0 1 2

±�X

0.8

0.5

X

pX

(0) = 0.2 pX

(100) = 0.8 · 0.5 pX

(300) = 0.8 · 0.5

E(X) = 0 · 0.2+ 100 · 0.8 · 0.5+ 300 · 0.8 · 0.5 = 160

pX

(0) = 0.5 pX

(200) = 0.5 · 0.2 pX

(300) = 0.5 · 0.8

E(X) = 0 · 0.5+ 200 · 0.5 · 0.2+ 300 · 0.5 · 0.8 = 140

p1

v1

p2

v2

E(X) = v1

p1

(1- p2

) + (v1

+ v2

)p1

p2

= p1

v1

+ p1

p2

v2

E(X) = v2

p2

(1- p1

) + (v1

+ v2

)p2

p1

= p2

v2

+ p1

p2

v1

p1

v1

+ p1

p2

v2

> p2

v2

+ p1

p2

v1

p1

v1

1- p1

> p2

v2

1- p2

pv

1- p

E(X) var(X)

X ⇠ U(a, b) a+b

2

(b-a)(b-a+2)12

X ⇠ Ber(p) p p(1- p)

X ⇠ Bin(n, p) np np(1- p)

X ⇠ Geo(p) 1

p

1-p

p

2

X ⇠ Poisson(�) � �

[a, b]

pX

(k) =

8><

>:

1

b- a+ 1k = a, a+ 1, . . . , b

0

c = b- a

E(X) =X

x

xpX

(x) = [(a) + (a+ 1) + . . .+ (a+ c)] · 1

c+ 1

=

(c+ 1)a+

c(c+ 1)

2

�· 1

c+ 1= a+

c

2=

a+ b

2

[1, n]

E�X2

�=

1

n

nX

k=1

k2 =1

6(n+ 1)(2n+ 1)

var(X) = E�X2

�- (E(X))2 =

1

6(n+ 1)(2n+ 1)-

1

4(n+ 1)2 =

n2 - 1

12

n = b- a+ 1

var(X) =(b- a+ 1)2 - 1

12=

(b- a)(b- a+ 2)

12

(p)

pX

(k) =

�p k = 1

1- p k = 0

E(X) =X

x

xpX

(x) = 1 · p+ 0 · (1- p) = p

E�X2

�=

X

x

x2 pX

(x) = 12 · p+ 02 · (1- p) = p

var(X) = E�X2

�-�E(X)

�2

= p- p2 = p(1- p)

�X

=pp(1- p)

(n, p)

pX

(k) = P(X = k) =

✓n

k

◆pk(1- p)n-k k = 0, 1, . . . , n

E(X) =nX

k=0

k

✓n

k

◆pk(1- p)n-k

=nX

k=1

kn!

k!(n- k)!pk(1- p)n-k

=nX

k=1

np(n- 1)!

(k- 1)![(n- 1)- (k- 1)]!pk-1(1- p)(n-1)-(k-1)

= np

E�X2

�=

nX

k=0

k2✓n

k

◆pk(1- p)n-k

= np

nX

k=1

k(n- 1)!

(k- 1)![(n- 1)- (k- 1)]!pk-1(1- p)(n-1)-(k-1)

= np

n-1X

k

0=0

(k 0 + 1)

✓n- 1

k 0

◆pk

0(1- p)(n-1)-(k 0) k 0 = k-1

= np

n-1X

k

0=0

(k 0)

✓n- 1

k 0

◆pk

0(1- p)(n-1)-(k 0)

!

+ np

= np · (n- 1)p+ np = np(1- p+ np)

var(X) = np(1- p+ np)- (np)2 = np(1- p)

(p)

pX

(k) = (1- p)k-1p k = 1, 2, . . .

S =1X

k=1

k(1- p)k-1

S = 1+ 2(1- p) + 3(1- p)2 + . . .

(1- p)S = (1- p) + 2(1- p)2 + . . .

pS = S- (1- p)S = 1+ (1- p) + (1- p)2 + . . . =1

1- (1- p)=

1

p

E(X) =1X

k=1

k(1- p)k-1p = pS =1

p

T = 1+ 4(1- p) + 9(1- p)2 + . . .

(1- p)T = (1- p) + 4(1- p)2 + . . .

T 0 = pT = 1+ 3(1- p) + 5(1- p)2 + . . .

(1- p)T 0 = (1- p) + 3(1- p)2 + . . .

pT 0 = 1+ 2(1- p) + 2(1- p)2 + . . . = 1+2(1- p)

p

E�X2

�=

1X

k=1

k2(1- p)k-1p = pT = T 0 =1

p+

2(1- p)

p2

var(X) =1

p+

2(1- p)

p2

-1

p2

=1- p

p2

(�)

pX

(k) =e-��k

k!k = 0, 1, . . .

E(X) =1X

k=0

ke-��k

k!=

1X

k=1

ke-��k

k!k = 0

= �

1X

k=1

e-��k-1

(k- 1)!m = k- 1

= �

1X

m=0

e-��m

m!

= �X

m

pX

(m) = �

E�X2

�=

1X

k=0

k2e-��k

k!

= �

1X

k=1

ke-��k-1

(k- 1)!

= �

1X

m=0

(m+ 1)e-��m

m!

= �(E(X) + 1) = �(�+ 1)

var(X) = E�X2

�-�E(X)

�2

= �(�+ 1)- �2 = �

X Y

pX,Y

(x, y) = P(X = x, Y = y)

X Y

pX

(x) =X

y

pX,Y

(x, y)

pY

(y) =X

x

pX,Y

(x, y)

X

x

X

y

pX,Y

(x, y) = 1

4 0 0.05 0.05 0.05

3 0.05 0.1 0.15 0.05

2 0.05 0.1 0.15 0.05

1 0.05 0.05 0.05 0

1 2 3 4

pX,Y

(x, y)y

x

0.15

0.35

0.35

0.15

pY

(y)

0.15 0.3 0.4 0.15

pX

(x)

Z = g(X, Y)

pZ

(z) =X

{(x,y) |g(x,y)=z}

pX,Y

(x, y)

E(g(X, Y)) =X

x

X

y

g(x, y)pX,Y

(x, y)

E(aX+ bY + c) = aE(X) + bE(Y) + c

X = X1

+ X2

+ . . .+ Xn

Xi

E(Xi

) = p

E(X) = E

nX

i=1

Xi

!

=nX

i=1

E(Xi

) = np

X A

pX |A(x) = P(X = x |A) =

P({X = x} \A)

P(A)

X

x

pX |A(x) = 1

X = x A

P(A)

pX

(k) = 1

6

k = 1, 2, 3, 4, 5, 6 A

pX |A(x)

pX |A(k) =

P({X = k} \ )

P( )

=

8<

:

1/6

1/2

= 1

3

k = 1, 3, 5

0

n

p

X

pX

(k) = (1- p)k-1p

A = {X 6 n}

P(A) =P

n

m=1

(1- p)m-1p

pX |A(k) =

8>>><

>>>:

(1- p)k-1pnP

m=1

(1- p)m-1p

k = 1, 2, . . . , n

0

A {Y = y}

X Y

pX |Y(x |y) = P(X = x | Y = y) =

pX,Y

(x, y)

pY

(y)

X

x

pX |Y(x |y) = 1

4 0 0.05 0.05 0.05

3 0.05 0.1 0.15 0.05

2 0.05 0.1 0.15 0.05

1 0.05 0.05 0.05 0

1 2 3 4

pX,Y

(x, y)y

x

pX |Y(x | 4) = {0, 1

3

, 13

, 13

}

pY |X(y | 2) = {1

6

, 13

, 13

, 16

}

pX,Y

(x, y) = pY

(y)pX |Y(x |y)

pX,Y

(x, y) = pX

(x)pY |X(y | x)

pX

(x) =X

y

pX,Y

(x, y) =X

y

pY

(y)pX |Y(x |y)

pY

(y) =X

x

pX,Y

(x, y) =X

x

pX

(x)pY |X(y | x)

X :

Y :

pY

(y) =

�5

6

y = 102

1

6

y = 104

pX |Y

�x | 102

�=

8>><

>>:

1

2

x = 10-2

1

3

x = 10-1

1

6

x = 1

pX |Y

�x | 104

�=

8>><

>>:

1

2

x = 11

3

x = 101

6

x = 100

pX

(x) =X

y

pY

(y)pX |Y(x |y)

=

8>>>>>>>>>><

>>>>>>>>>>:

5

6

· 1

2

x = 10-2

5

6

· 1

3

x = 10-1

5

6

· 1

6

+ 1

6

· 1

2

x = 1

1

6

· 1

3

x = 10

1

6

· 1

6

x = 102

B

E(X |B) =nX

i=1

P(Ai

|B)E(X |Ai

\ B)

Ai

= {Y = y}

E(X) =X

y

pY

(y)E(X | Y = y)

0.5 0.05

0.2 0.3

0.3 0.1

E(X) = 0.5 · 0.05+ 0.2 · 0.3+ 0.3 · 0.1 = 0.115

pX

(k) = (1-p)k-1p

�A

1

= {X = 1} = { }

A2

= {X > 1} = { }

E(X) = P(A1

)E(X |A1

) + P(A2

)E(X |A2

)

= p · 1+ (1- p) · (E(X) + 1)

E(X) = 1

p

E�X2

�= P(A

1

)E�X2 |A

1

�+ P(A

2

)E�X2 |A

2

�

= p · 1+ (1- p) · E�(X+ 1)2

�

= p+ (1- p)(1+ 2E(X) + E�X2

�)

E�X2

�= 2

p

2

- 1

p

var(X) = 1-p

p

2

pX

(k) = (1-p)k-1p�A

1

= {X = 1} = { }

A2

= {X > 1} = { }

E(X) = P(A1

)E(X |A1

) + P(A2

)E(X |A2

)

= p · 1+ (1- p) · (E(X) + 1)

E(X) = 1

p

E�X2

�= P(A

1

)E�X2 |A

1

�+ P(A

2

)E�X2 |A

2

�

= p · 1+ (1- p) · E�(X+ 1)2

�

= p+ (1- p)(1+ 2E(X) + E�X2

�)

E�X2

�= 2

p

2

- 1

p

var(X) = 1-p

p

2

{X = x} A X

A

P({X = x} \A) = P(X = x)P(A) = pX

(x)P(A)

pX |A(x) =

P({X = x} \A)

P(A)

pX |A(x) = p

X

(x)

{X = x} {Y = y} X

Y

pX,Y

(x, y) = pX

(x)pY

(y)

pX,Y

(x, y) = pX |Y(x |y)pY

(y) y pY

(y) > 0

pX |Y(x |y) = p

X

(x)

X Y

A = {X 6 2 \ Y > 3}

4 0.05 0.1 0.1 0

3 0.1 0.2 0.05 0.1

2 0 0.05 0.15 0.05

1 0 0.05 0 0

1 2 3 4

pX,Y

(x, y)y

x

pX |A(x) =

�1

3

x = 12

3

x = 2

pY |A(y) =

�1

3

y = 42

3

y = 3

pX,Y |A(x, y) =

8>>>><

>>>>:

1

9

x = 1, y = 42

9

x = 1, y = 32

9

x = 2, y = 44

9

x = 2, y = 3

X Y

A = { }

pX |A(x) =

�1

2

x = 01

2

x = 1

pY |A(x) =

�1

2

y = 01

2

y = 1

pX,Y |A(x) =

�1

2

x = 0, y = 11

2

x = 1, y = 0

pX,Y |A(x, y) 6= p

X |A(x)pY |A(y)

X Y

E(XY) = E(X)E(Y)E(g(X)h(Y)) = E(g(X))E(h(Y))

E(XY) =X

x

X

y

xypX,Y

(x, y)

=X

x

X

y

xypX

(x)pY

(y)

=X

x

xpX

(x)X

y

ypY

(y) = E(X)E(Y)

X Y

var(X+ Y) = var(X) + var(Y)

˜X = X- E(X) ˜Y = Y - E(Y) E⇣˜X+ ˜Y

⌘= 0

E⇣˜X˜Y⌘= E

⇣˜X⌘E⇣˜Y⌘= 0

var(X+ Y) = var

⇣˜X+ ˜Y

⌘= E

⇣(˜X+ ˜Y)2

⌘-⇣E⇣˜X+ ˜Y

⌘⌘2

= E⇣˜X2 + 2˜X˜Y + ˜Y2

⌘

= E⇣˜X2

⌘+ 2E

⇣˜X˜Y⌘+ E

⇣˜Y2

⌘

= E⇣˜X2

⌘+ E

⇣˜Y2

⌘

= var

⇣˜X⌘+ var

⇣˜Y⌘= var(X) + var(Y)

X = X1

+ X2

+ . . .+ Xn

Xi

E(Xi

) = p

E(X) = np

var(X) = var

nX

i=1

Xi

!

=nX

i=1

var(Xi

) = np(1- p)

X

MX

(s) M(s)

MX

(s) = E�esX�

M(s) =X

x

esxpX

(x)

d

dsM(s) =

d

dsE�esX�= E

✓d

dsesX◆

= E�XesX

�

E(X) = d

dsM(s)

��s=0

E�X2

�=

d

2

ds2M(s)

��s=0

M(s) E(X) E�X2

�var(X) p

X

(x) =

8>><

>>:

1

2

x = 21

6

x = 31

3

x = 5

M(s) = E�esX�= 1

2

e2s + 1

6

e3s + 1

3

e5s

E(X) = d

dsM(s)

��s=0

= 1

2

· 2e2s + 1

6

· 3e3s + 1

3

· 5e5s��s=0

= 19

6

E�X2

�=

d

2

ds2M(s)

��s=0

= 1

2

· 4e2s + 1

6

· 9e3s + 1

3

· 25e5s��s=0

= 71

6

var(X) = 71

6

- (196

)2 = 65

36

[a, b]

pX

(k) =

8><

>:

1

b- a+ 1k = a, a+ 1, . . . , b

0

M(s) = E�esX�=

bX

x=a

esx1

b- a+ 1

=esa

b- a+ 1

⇣1+ es + e2s + . . .+ e(b-a)s

⌘

=esa

b- a+ 1· e

(b-a+1)s - 1

es - 1

(p)

pX

(k) =

�p k = 1

1- p k = 0

M(s) =X

x

esx pX

(x) = es · p+ e0 · (1- p) = 1- p+ pes

E(X) = d

ds(1- p+ pes)

��s=0

= p

E�X2

�=

d

2

ds2(1- p+ pes)

��s=0

= p

X Y Z = X+ Y

MZ

(s) = MX

(s)MY

(s)

MZ

(s) = E�esZ�= E

⇣es(X+Y)

⌘= E

�esXesY

�

X Y

E�esXesY

�= E

�esX�E�esY�= M

X

(s)MY

(s)

N

X = X1

+ X2

+ . . .+ Xn

Xi

E(Xi

) = p

MX

i

(s) = 1- p+ pes

MX

(s) = (1- p+ pes)n

E(X) = d

ds(1- p+ pes)n

��s=0

= np

E�X2

�=

d

2

ds2(1- p+ pes)n

��s=0

= np(1- p+ np)

(p)

pX

(k) = (1- p)k-1p k = 1, 2, . . .

MX

(s) =1X

x=1

(1- p)x-1pesx

= pes1X

x=1

[(1- p)es]x-1

=pes

1- (1- p)es

(�)

pX

(k) =e-��k

k!k = 0, 1, . . .

M(s) =1X

x=0

esx�xe-�

x!= e-�

1X

x=0

(es�)x

x!= e-�ee

s

� = e�(es-1)

E(X) = e�(es-1) · �es

��s=0

= �

E�X2

�=⇥e�(e

s-1) · (�es)2 + e�(es-1) · �es

⇤s=0

= �2 + �

X Y � µ

Z = X+ Y pZ

(z)

MX

(s) = e�(es-1) M

Y

(s) = eµ(es-1)

MZ

(s) = MX

(s)MY

(s)

= e�(es-1)eµ(es-1)

= e(�+µ)(es-1)

Z ⇠ Poisson(�+ µ)

discrete random variables - university of hong kongelec2844/sp21/lec2.pdfx px(0) = 0.2 px(100) =...

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