� Venn diagram : 1 . Region I : A�
B 2 . Region II : A B� � 3 . Region III : A��
B 4 . Region IV : ( A�
B) �
� Tree diagram :
1 . Path I : P ( A�B) � pq
2 . Path I + Path III : �
P ( B)
� P ( A� B) � P ( A� � B)
� pq � ( 1 � p) r
� Markov Chain with transition matrix T : 1 . det ( T��I ) : Characteristic polynomial of T 2 . Solution ( s ) of det ( T��I ) �0
: Eigenvalue ( s ) of T 3 . v : Steady state probability vector , which is the eigenvector of T corresponding to the eigenvalue � � 1 4 . v
0
: Initial state probability vector 5 . v � n n
T v
0
: State probability vector after n transitions 6 . The column sum of T must be equal to 1
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Discrete Probability Distributions
� Properties of a discrete random variable X :
X x
1 x
2 … x n
P ( X x)
� P ( X � x1
) X x2 1 . P ( X � x1) � P ( X � x2) � � P ( X � x n
) � 1
P ( � ) … P ( X � x n )
2 . E ( X ) � x1P ( X � x1 ) � x2P ( X � x2) � � xnP ( X � xn) : Expected value of X
3 . E ( X ) � 0 if a fair game is considered
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