Discrete Random Variables - University of Hong Kongelec2844/sp21/lec2.pdfX pX(0) = 0.2 pX(100) =...
Transcript of Discrete Random Variables - University of Hong Kongelec2844/sp21/lec2.pdfX pX(0) = 0.2 pX(100) =...
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
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
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
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
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
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
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
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
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
X Y A
E(X |A) =X
x
xpX |A(x)
E(g(X) |A) =X
x
g(x)pX |A(x)
E(X | Y) =X
x
xpX |Y(x |y)
E(g(X) | Y) =X
x
g(x)pX |Y(x |y)
A1
, . . . , An
P(Ai
) > 0 i
E(X) =nX
i=1
P(Ai
)E(X |Ai
)
=
pX
(x) =nP
i=1
P(Ai
)pX |A
i
(x)
E(X) =X
x
xpX
(x) =X
x
x
nX
i=1
P(Ai
)pX |A
i
(x) =nX
i=1
P(Ai
)X
x
xpX |A
i
(x)
=nX
i=1
P(Ai
)E(X |Ai
)
A1
, . . . , An
P(Ai
) > 0 i
E(X) =nX
i=1
P(Ai
)E(X |Ai
)
=
pX
(x) =nP
i=1
P(Ai
)pX |A
i
(x)
E(X) =X
x
xpX
(x) =X
x
x
nX
i=1
P(Ai
)pX |A
i
(x) =nX
i=1
P(Ai
)X
x
xpX |A
i
(x)
=nX
i=1
P(Ai
)E(X |Ai
)
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
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
P(X = x, Y = y |A) = P(X = x |A)P(Y = y |A)
pX,Y |A(x, y) = p
X |A(x)pY |A(y)
pX |Y,A(x |y) = p
X |A(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
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
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 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
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)
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)