Formulas
Axiom 1 For any event A, 0 ≤ P [A].
Axiom 2 P [S] = 1.
Axiom 3 For any countable collection A1 , A2 , . . . of mutually exclusive events
P [A1 ∪ A2 ∪ . . .] = P [A1 ] + P [A2 ] + . . . .
P [AC ] = 1 − P [A]
(A ∪ B)C = AC ∩ B C
(A ∩ B)C = AC ∪ B C
If (A ∩ B) = φ A and B are disjoint.
P [A|B] P [B]
P [A ∪ B] = P [A] + P [B] − P [A ∩ B]
P [A ∩ B] = P [A|B]P [B]
P [B|A] =
P [A]
Events A and B are said to be independent iff P [A ∩ B] = P [A] P [B].
n X
n!
n!
n
n
n
n
=
=
(p + q) =
pk q n−k
n0 n1 . . . nm−1
k
k
k!(n − k)!
n0 !n1 ! . . . nm−1 !
k=0
Table 1. The Bernoulli family of PMF’s
Name
PMF


1 − p x = 0
Bernoulli
PX [x] = p
x=1


0
otherwise
!


 n pk (1 − p)n−k k = 0, 1, . . . , n
Binomial PK [k] =
k


0
otherwise
(
y−1
p(1 − p)
y = 1, 2, . . .
Geometric
PY [y] =
0
otherwise
(
l−1 k
l−k
l = k, k + 1, . . .
k−1 p (1 − p)
Pascal
PL [l] =
0
otherwise
(
αn e−α /n! n = 0, 1, 2, . . .
Poisson
PN [n] =
0
otherwise
∞
X
p=0
E[g(X)] =
X
1
p =
1−p
n
pi g(xi )
x
e =
∞
X
xn
n=0
n!
n
X
k=1
n(n + 1)
k=
2
E[aX1 + bX2 ] = aE[X1 ] + bE[X2 ]
n
X
k2 =
k=1
Var[X]
p
p(1 − p)
np
np(1 − p)
1/p
(1 − p)/p2
k/p
k(1 − p)/p2
α
α
n(n + 1)(2n + 1)
6
2
Var[aX] = a2 E[(X − µX )2 ] = a2 Var[X] = a2 σX
i
(CDF) FX [xk ] = P [X ≤ xk ] =
X
j≤k
1
E[X]
P [xj ]