Summary

Properties
DISTRIBUTIONS
P(x) 1 =




Ifdisjoint, P(AUB) P(A) P(B) =
+
1. Bernouli:1 -Ber(p) ifp(1=1) P =




p( =0) 1
=



P
-




P(A2) 1 =
-

P(A) E(X)
·
=




P
A
r B Both A
a nd B
=




V(x) 4)
·


p. (1
=

-




AUB=A ORB is enough


De 2. Binomial:EiBer(p) independently i 1, ...,
Morgan's laws across
=




AUB)=A BE
A B
I "E i
=




(ArB)* A UBS
=
·

E(I) np=



-Bin(n,p) n'de successes en ntrials


P(A1B) P(AUBC V(x) xp. (1 p)
=



1
·


R!
=
-
-


n =




* K
k!(n k)!
4.p". (1-p)"
-




P(AUB)
-




P(A) P(B) P(AB) PMF:P(1=K)
·
=
- =
+




R! 1.
=
(1-1)....
CONDITIONAL PROBABILITY

xx
Bayes:P(AB) P(AB)
-




P(A). P(BIA) 3. Poisson:Poisson (x) ifP(1=K) e
=


PMF
· = =
=




P(B) P(B) k!
·

E(E) V(E) X.
=
=

Paracuando unknown upperlimit
·
Productr ule:P(AB1C...) P(A). P(BA). P(C/A,B)... =




4. Geometric:P(1=K) (1-p) .p.
-




=

Stop when success
LAW OFTOTAL PROBABILITY
similar to Bin, butto know total n of trials before success
P(A) [P(A1Bi) [P(Bi).P(A
=
=

BiS

DISCRETERANDOM VARIABLES

PMF of :P(x) P(A x) = -



x
· =




·
CDF 1:
of F(x) P(1-x) 2P(X)
= =




y x


·
Mean / Expectation:E(E) 2 xiP(Xi) =




Properties:if2 a.x b
=
+




1. E(z) a.E(A) b.E(Y)
=
+




2. E(X =I) E(A) =E(Y)
=




3. E(a +b) a.E(A) b =
+




·Variance:V(I):[(xi-E(E)]2.P(xi)

Properties
1. V(ax) a2.V(E)
=




2. Ifindependent:V(A + y) V(E x) V(x) v(I)
=
-
=
+




2
3. V(E) E(12)
=
-

[E(E)]

4. V!a b) V(I) +
=




·

3D:SD(A) v(x) =




IndependentEvents

P(AB) P(A) < 0
P(AB) P(A). P(B)
=


=