Probability Theory for EOR
(University of Groningen)
Summary 2021-2022
Stuvia: marcellaschrijver
Probability and counting ........................................................................................................................ 2
Conditional probability .......................................................................................................................... 3
Expectation ............................................................................................................................................. 4
Random variables ................................................................................................................................... 4
Overview ............................................................................................................................................. 4
Discrete ............................................................................................................................................... 5
Continuous .......................................................................................................................................... 6
Moments ................................................................................................................................................ 7
Distributions ........................................................................................................................................... 9
, Probability and counting
Naive definition of probability
|𝐴| 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑡𝑜 𝐴
𝑃𝑛𝑎𝑖𝑣𝑒 (𝐴) = |𝑆|
= 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑆
o Equally likely outcomes
o Finite 𝑆
How to count
Binomial
𝑛 𝑛(𝑛 − 1) … (𝑛 − 𝑘 + 1) 𝑛!
( )= =
𝑘 𝑘! (𝑛 − 𝑘)! 𝑘!
𝑛
𝑛
∑ ( ) 𝑝𝑘 𝑞𝑛−𝑘 = (𝑝 + 𝑞)𝑛
𝑘
𝑘=0
Properties of probability
𝑃(∅) = 0, 𝑃(𝑆) = 1
If 𝐴1 , 𝐴2 , … are disjoint (𝐴𝑖 ∩ 𝐴𝑗 = ∅ for 𝑖 ≠ 𝑗) ⟹ 𝑃(⋃∞ ∞
𝑛=1 𝐴𝑛 ) = ∑𝑛=1 𝑃(𝐴𝑛 )
𝑃(𝐴𝑐 ) = 1 − 𝑃(𝐴)
𝐴 implies 𝐵: 𝐴 ⊆ 𝐵 ⟹ 𝑃(𝐴) ≤ 𝑃(𝐵)
Inclusion-exclusion: 𝑃(⋃𝑛𝑖=1 𝐴𝑖 ) = ∑𝑖 𝑃(𝐴𝑖 ) − ∑𝑖<𝑗 𝑃(𝐴𝑖 ∩ 𝐴𝑗 ) + ∑𝑖<𝑗<𝑘 𝑃(𝐴𝑖 ∩ 𝐴𝑗 ∩ 𝐴𝑘 ) − ⋯ +
(−1)𝑛+1 𝑃(𝐴𝑖 ∩ … ∩ 𝐴𝑛 )
Vandermonde’s identity
𝑘
𝑚+𝑛 𝑚 𝑛
( ) = ∑( )( )
𝑘 𝑗 𝑘−𝑗
𝑗=0
(University of Groningen)
Summary 2021-2022
Stuvia: marcellaschrijver
Probability and counting ........................................................................................................................ 2
Conditional probability .......................................................................................................................... 3
Expectation ............................................................................................................................................. 4
Random variables ................................................................................................................................... 4
Overview ............................................................................................................................................. 4
Discrete ............................................................................................................................................... 5
Continuous .......................................................................................................................................... 6
Moments ................................................................................................................................................ 7
Distributions ........................................................................................................................................... 9
, Probability and counting
Naive definition of probability
|𝐴| 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑡𝑜 𝐴
𝑃𝑛𝑎𝑖𝑣𝑒 (𝐴) = |𝑆|
= 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑆
o Equally likely outcomes
o Finite 𝑆
How to count
Binomial
𝑛 𝑛(𝑛 − 1) … (𝑛 − 𝑘 + 1) 𝑛!
( )= =
𝑘 𝑘! (𝑛 − 𝑘)! 𝑘!
𝑛
𝑛
∑ ( ) 𝑝𝑘 𝑞𝑛−𝑘 = (𝑝 + 𝑞)𝑛
𝑘
𝑘=0
Properties of probability
𝑃(∅) = 0, 𝑃(𝑆) = 1
If 𝐴1 , 𝐴2 , … are disjoint (𝐴𝑖 ∩ 𝐴𝑗 = ∅ for 𝑖 ≠ 𝑗) ⟹ 𝑃(⋃∞ ∞
𝑛=1 𝐴𝑛 ) = ∑𝑛=1 𝑃(𝐴𝑛 )
𝑃(𝐴𝑐 ) = 1 − 𝑃(𝐴)
𝐴 implies 𝐵: 𝐴 ⊆ 𝐵 ⟹ 𝑃(𝐴) ≤ 𝑃(𝐵)
Inclusion-exclusion: 𝑃(⋃𝑛𝑖=1 𝐴𝑖 ) = ∑𝑖 𝑃(𝐴𝑖 ) − ∑𝑖<𝑗 𝑃(𝐴𝑖 ∩ 𝐴𝑗 ) + ∑𝑖<𝑗<𝑘 𝑃(𝐴𝑖 ∩ 𝐴𝑗 ∩ 𝐴𝑘 ) − ⋯ +
(−1)𝑛+1 𝑃(𝐴𝑖 ∩ … ∩ 𝐴𝑛 )
Vandermonde’s identity
𝑘
𝑚+𝑛 𝑚 𝑛
( ) = ∑( )( )
𝑘 𝑗 𝑘−𝑗
𝑗=0
