Exam P

Adapt to Your Exam

General Probability

Basic Probability Relationships De Morgan’s Law Bayes’ Theorem

General Probability GENERAL
-]
= Pr(𝐴𝐴- ∩ PROBABILITY
𝐵𝐵 -) Pr(𝐵𝐵|𝐴𝐴8 ) ⋅ Pr(𝐴𝐴8 )
Pr(𝐴𝐴 ∪ 𝐵𝐵) = Pr(𝐴𝐴) + Pr(𝐵𝐵) − Pr(𝐴𝐴 ∩ 𝐵𝐵) Pr[(𝐴𝐴 ∪ 𝐵𝐵)
Pr[(𝐴𝐴 ∩ 𝐵𝐵) = Pr(𝐴𝐴 ∪ 𝐵𝐵 -)
-] -

Pr(𝐴𝐴8 |𝐵𝐵) = 1
Pr(𝐴𝐴 ∪ 𝐵𝐵 ∪ 𝐶𝐶) = Pr(𝐴𝐴) + Pr(𝐵𝐵) + Pr(𝐶𝐶)
Basic Probability Relationships De Morgan’s Law Bayes’ Theorem ∑023 Pr(𝐵𝐵|𝐴𝐴0 ) ⋅ Pr(𝐴𝐴0 )

− Pr(𝐴𝐴
Pr(𝐴𝐴 ∪ 𝐵𝐵) = Pr(𝐴𝐴) + Pr(𝐵𝐵) ∩ 𝐵𝐵) ∩
− Pr(𝐴𝐴 − 𝐵𝐵)
Pr(𝐵𝐵 ∩ 𝐶𝐶) Pr[(𝐴𝐴 ∪ 𝐵𝐵)-] = Pr(𝐴𝐴- ∩ 𝐵𝐵 -) Pr(𝐵𝐵|𝐴𝐴8 ) ⋅ Pr(𝐴𝐴8 )
Conditional Probability Pr(𝐴𝐴 8 |𝐵𝐵) = 1
Combinatorics
Pr(𝐴𝐴 ∪ 𝐵𝐵 ∪ 𝐶𝐶) = Pr(𝐴𝐴) −
+Pr(𝐴𝐴
Pr(𝐵𝐵)∩+𝐶𝐶)Pr(𝐶𝐶)
+ Pr(𝐴𝐴 ∩ 𝐵𝐵 ∩ 𝐶𝐶) Pr[(𝐴𝐴 ∩ 𝐵𝐵)-] = Pr(𝐴𝐴
Pr(𝐴𝐴
-
∪ 𝐵𝐵 -)
∩ 𝐵𝐵) ∑023 Pr(𝐵𝐵|𝐴𝐴0 ) ⋅ Pr(𝐴𝐴0 )
Pr(𝐴𝐴- ) = 1 − Pr(𝐴𝐴) − Pr(𝐴𝐴 ∩ 𝐵𝐵) − Pr(𝐵𝐵 ∩ 𝐶𝐶) 𝑛𝑛! = 𝑛𝑛 ⋅ (𝑛𝑛 − 1) ⋅ … ⋅ 2 ⋅ 1

Pr(𝐴𝐴|𝐵𝐵) =
Conditional Probability
Pr(𝐵𝐵) Combinatorics 𝑛𝑛!
− Pr(𝐴𝐴 ∩ 𝐶𝐶) + Pr(𝐴𝐴 ∩ 𝐵𝐵 ∩ 𝐶𝐶)
Law of Total Probability
Pr(𝐴𝐴 ∩ 𝐵𝐵) 𝑃𝑃 =
Pr(𝐴𝐴- ) = 1 −1 Pr(𝐴𝐴) Independence
Pr(𝐴𝐴|𝐵𝐵) = 𝑛𝑛! 1= 8𝑛𝑛 ⋅ (𝑛𝑛
(𝑛𝑛−−1) 𝑘𝑘)!⋅ … ⋅ 2 ⋅ 1

Pr(𝐴𝐴 ∩ 𝐵𝐵) = Pr(𝐵𝐵)
Pr(𝐴𝐴) ⋅ Pr(𝐵𝐵) 𝑛𝑛!
Pr(𝐵𝐵) = / Pr(𝐵𝐵 ∩ 𝐴𝐴0 ) 𝑛𝑛 𝑛𝑛!
Law of Total Probability
𝑃𝑃 =
8𝐶𝐶8 = @ A =
1 023 Pr(𝐴𝐴|𝐵𝐵) = Pr(𝐴𝐴)
Independence 1 1
(𝑛𝑛 −𝑘𝑘 𝑘𝑘)! (𝑛𝑛 − 𝑘𝑘)! ⋅ 𝑘𝑘!
Pr(𝐵𝐵) = / Pr(𝐵𝐵 ∩ 𝐴𝐴0 ) ∩ 𝐵𝐵) = Pr(𝐴𝐴) ⋅ Pr(𝐵𝐵)
Pr(𝐴𝐴 𝑛𝑛 𝑛𝑛!
Pr(𝐴𝐴|𝐵𝐵)
= Pr(𝐴𝐴) 1𝐶𝐶
8 = @𝑘𝑘 A = (𝑛𝑛 − 𝑘𝑘)! ⋅ 𝑘𝑘!
023
Univariate Probability Distributions

Variance, Standard Deviation, and Probability Generating Function (PGF)
*Probability Mass Function (PMF)
Univariate Probability Coefficient of Variation 𝑃𝑃 (𝑡𝑡) = 𝐸𝐸[𝑡𝑡 C ]

∑EFF H 𝑝𝑝C (𝑥𝑥) = 1 Distributions UNIVARIATE

PROBABILITY DISTRIBUTIONS
C
Pr(𝑋𝑋 = 𝑎𝑎) = 0 (continuous)
*Probability Mass Function (PMF) 𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋] = 𝐸𝐸[𝑋𝑋 c ] − (𝐸𝐸[𝑋𝑋])c
Variance, Standard Deviation, and 𝑃𝑃C (0) = 𝑝𝑝C (0)
Probability Generating Function (PGF)

∑EFF H 𝑝𝑝C (𝑥𝑥) = 1 𝑉𝑉𝑉𝑉𝑉𝑉[𝑎𝑎𝑎𝑎 + 𝑏𝑏] = 𝑎𝑎c ⋅ 𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋]
Coefficient of Variation 𝑑𝑑1 = 𝐸𝐸[𝑡𝑡 C ]
𝑃𝑃C (𝑡𝑡)
*Cumulative Distribution Function (CDF) 1 𝑃𝑃C (𝑡𝑡)o
Pr(𝑋𝑋 = 𝑎𝑎) = 0 (continuous) 𝑉𝑉𝑉𝑉𝑉𝑉[𝑐𝑐]==𝐸𝐸[𝑋𝑋
𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋] 0 c ] − (𝐸𝐸[𝑋𝑋])c 𝑃𝑃C𝑑𝑑𝑡𝑡(0) = 𝑝𝑝C (0) j2\
= 𝑝𝑝C (𝑛𝑛)
𝐹𝐹 (𝑥𝑥) = Pr(𝑋𝑋 ≤ 𝑥𝑥) = ∑ 𝑝𝑝 (𝑖𝑖)
C 0OH C
𝑉𝑉𝑉𝑉𝑉𝑉[𝑎𝑎𝑎𝑎
𝑆𝑆𝑆𝑆[𝑋𝑋]+=𝑏𝑏] = 𝑎𝑎c ⋅ 𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋]
e𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋] 𝑑𝑑1 𝑛𝑛!
Pr(𝑎𝑎 < 𝑋𝑋 ≤ 𝑏𝑏) = 𝐹𝐹C (𝑏𝑏) − 𝐹𝐹C (𝑎𝑎)
*Cumulative Distribution Function (CDF) 𝑃𝑃
1 (𝑡𝑡)o
𝑉𝑉𝑉𝑉𝑉𝑉[𝑐𝑐]
𝐶𝐶𝐶𝐶[𝑋𝑋]==0 𝑆𝑆𝑆𝑆[𝑋𝑋]⁄𝐸𝐸[𝑋𝑋] 𝑑𝑑𝑡𝑡 1𝑑𝑑 C
𝐹𝐹C𝑓𝑓(𝑥𝑥)
S
𝑥𝑥) = ∑0OH 𝑝𝑝C (𝑖𝑖)
= =Pr(𝑋𝑋𝐹𝐹C≤(𝑥𝑥) (continuous) 𝑃𝑃 (𝑡𝑡)o
j2\
== (𝑛𝑛) − 1) … (𝑋𝑋 − 𝑛𝑛 + 1)]
𝑝𝑝C𝐸𝐸[𝑋𝑋(𝑋𝑋
C (𝑥𝑥)
𝑑𝑑𝑡𝑡 1𝑛𝑛!C
SH
Pr(𝑎𝑎 < 𝑋𝑋 ≤ 𝑏𝑏) = 𝐹𝐹C (𝑏𝑏) − 𝐹𝐹C (𝑎𝑎) 𝑆𝑆𝑆𝑆[𝑋𝑋] = e𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋]
*Moment Generating Function (MGF) 1
j23

𝐶𝐶𝐶𝐶[𝑋𝑋] == 𝑆𝑆𝑆𝑆[𝑋𝑋] ⁄ ] 𝑑𝑑

𝑓𝑓C *Expected Value

S
(𝑥𝑥) = 𝐹𝐹C (𝑥𝑥) (continuous) 𝑀𝑀 (𝑡𝑡)C 𝐸𝐸[𝑒𝑒 jC𝐸𝐸[𝑋𝑋] Percentiles
𝑃𝑃 (𝑡𝑡)o
1 C
= 𝐸𝐸[𝑋𝑋(𝑋𝑋 − 1) … (𝑋𝑋 − 𝑛𝑛 + 1)]

𝑑𝑑𝑡𝑡The 100𝑝𝑝
𝐸𝐸[𝑐𝑐] = 𝑐𝑐
SH th percentile is the smallest value of 𝜋𝜋
j23
Y
𝑀𝑀kClm (𝑡𝑡) = 𝑒𝑒 mj ⋅ 𝑀𝑀C (𝑎𝑎𝑎𝑎)
*Moment Generating Function (MGF) q
𝐸𝐸[𝑔𝑔(𝑋𝑋)] = ∫ZY 𝑔𝑔(𝑥𝑥) ⋅ 𝑓𝑓C (𝑥𝑥) 𝑑𝑑𝑑𝑑
*Expected Value C (0)
𝑀𝑀C𝑀𝑀(𝑡𝑡) == 1 jC ]
𝐸𝐸[𝑒𝑒 Percentiles
where 𝐹𝐹C r𝜋𝜋qs ≥ 𝑝𝑝.
𝐸𝐸[𝑐𝑐] = 𝑐𝑐 = ∫Y 𝑔𝑔- (𝑥𝑥) ⋅ 𝑆𝑆 (𝑥𝑥) 𝑑𝑑𝑑𝑑, for domain 𝑥𝑥 ≥ 0
𝐸𝐸[𝑔𝑔(𝑋𝑋)] 𝑀𝑀Cln(𝑡𝑡)
𝑀𝑀kClm (𝑡𝑡)==𝑒𝑒𝑀𝑀
mj (𝑡𝑡) ⋅(𝑎𝑎𝑎𝑎)
C⋅ 𝑀𝑀C 𝑀𝑀n (𝑡𝑡) (independent) The 100𝑝𝑝
th percentile is the smallest value of 𝜋𝜋
q
Y\ C 1 Univariate Transformation
𝐸𝐸[𝑔𝑔(𝑋𝑋)] = ∫ZY 𝑔𝑔(𝑥𝑥) ⋅ 𝑓𝑓C (𝑥𝑥) 𝑑𝑑𝑑𝑑 8 𝑀𝑀C𝑑𝑑(0) = (𝑡𝑡)o1 where 𝐹𝐹C r𝜋𝜋qs ≥ 𝑝𝑝. 𝑑𝑑
∫_ 𝑔𝑔(𝑥𝑥) ⋅ 𝑓𝑓C (𝑥𝑥) 𝑑𝑑𝑑𝑑 𝑀𝑀 = 𝐸𝐸[𝑋𝑋 1 ]
𝐸𝐸[𝑔𝑔(𝑋𝑋)] =
𝐸𝐸[𝑔𝑔(𝑋𝑋)|𝑗𝑗 ≤∫
Y -
𝑔𝑔 (𝑥𝑥) ⋅ 𝑆𝑆 (𝑥𝑥) 𝑑𝑑𝑑𝑑 , for domain 𝑥𝑥 ≥ 0 𝑑𝑑𝑡𝑡 1(𝑡𝑡)C= 𝑀𝑀j2\
𝑀𝑀Cln C (𝑡𝑡) ⋅ 𝑀𝑀n (𝑡𝑡) (independent)

𝑓𝑓n (𝑦𝑦) = 𝑓𝑓C [𝑔𝑔Z3(𝑦𝑦)] ⋅ o 𝑔𝑔Z3 (𝑦𝑦)o
\ 𝑋𝑋 ≤ 𝑘𝑘] =C 1 Univariate Transformation 𝑑𝑑𝑑𝑑
8 Pr(𝑗𝑗 ≤ 𝑋𝑋 ≤ 𝑘𝑘) 𝑑𝑑

∫_ 𝑔𝑔(𝑥𝑥) ⋅ 𝑓𝑓C (𝑥𝑥) 𝑑𝑑𝑑𝑑

𝐸𝐸[𝑐𝑐 ⋅ 𝑔𝑔(𝑋𝑋)] = 𝑐𝑐 ⋅ 𝐸𝐸[𝑔𝑔(𝑋𝑋)] 𝑀𝑀 (𝑡𝑡)o = 𝐸𝐸[𝑋𝑋 1 ] where 𝑦𝑦 = Z3 𝑔𝑔(𝑥𝑥) ⇔ 𝑥𝑥𝑑𝑑 = Z3 𝑔𝑔Z3(𝑦𝑦)
𝐸𝐸[𝑔𝑔(𝑋𝑋)|𝑗𝑗 ≤ 𝑋𝑋 ≤ 𝑘𝑘] = 𝑑𝑑𝑡𝑡 1 C j2\ 𝑓𝑓n (𝑦𝑦) = 𝑓𝑓C [𝑔𝑔 (𝑦𝑦)] ⋅ o 𝑔𝑔 (𝑦𝑦)o
𝐸𝐸[𝑔𝑔3 (𝑋𝑋) + ⋯ + 𝑔𝑔8 (𝑋𝑋)]Pr(𝑗𝑗 ≤ 𝑋𝑋3 (𝑋𝑋)]
= 𝐸𝐸[𝑔𝑔 ≤ 𝑘𝑘)+ ⋯ + 𝐸𝐸[𝑔𝑔8 (𝑋𝑋)] 𝑑𝑑𝑑𝑑

𝐸𝐸[𝑐𝑐
⋅ 𝑔𝑔(𝑋𝑋)] = 𝑐𝑐 ⋅ 𝐸𝐸[𝑔𝑔(𝑋𝑋)]
where 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) ⇔ 𝑥𝑥 = 𝑔𝑔Z3(𝑦𝑦)
3 (𝑋𝑋) + ⋯ + 𝑔𝑔8 (𝑋𝑋)] = 𝐸𝐸[𝑔𝑔3 (𝑋𝑋)] + ⋯ + 𝐸𝐸[𝑔𝑔8 (𝑋𝑋)]
𝐸𝐸[𝑔𝑔

Discrete Distributions

PMF Mean Variance MGF PGF Special Properties
Discrete Distributions 1 𝑎𝑎 + 𝑏𝑏 (𝑏𝑏 − 𝑎𝑎 + 1)c − 1 𝑒𝑒 kj − 𝑒𝑒 (ml3)j
Discrete Uniform
PMF Mean Variance MGF –
PGF –
Special Properties
𝑏𝑏 − 𝑎𝑎 + 1 2 12 (1 − 𝑒𝑒 j )(𝑏𝑏 − 𝑎𝑎 + 1)
1 𝑎𝑎 + 𝑏𝑏 (𝑏𝑏 − 𝑎𝑎 + 1)c − 1 𝑒𝑒 kj − 𝑒𝑒 (ml3)j
Discrete Uniform 𝑛𝑛 H – –
Binomial @ 𝑏𝑏A−𝑝𝑝 𝑎𝑎(1+−1 𝑝𝑝) 1ZH 2𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛(1
12 − 𝑝𝑝) )(𝑏𝑏j−+𝑎𝑎𝑞𝑞)+1 1)
(1 − 𝑒𝑒 j(𝑝𝑝𝑒𝑒 (𝑝𝑝𝑝𝑝 + 𝑞𝑞)1 –
𝑥𝑥
𝑛𝑛 𝑚𝑚
Binomial @ 𝑚𝑚A A𝑝𝑝H⋅ (1
w@
𝑁𝑁−−𝑝𝑝)
𝑚𝑚1ZH 𝑁𝑁 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛(1 −– 𝑝𝑝) (𝑝𝑝𝑒𝑒 j + –
𝑞𝑞)1 (𝑝𝑝𝑝𝑝 + 𝑞𝑞)
–
1
– –
Hypergeometric 𝑥𝑥𝑥𝑥 @ 𝑛𝑛 − 𝑥𝑥 Az{@ 𝑛𝑛 A 𝑛𝑛 ⋅
𝑁𝑁
𝑚𝑚 𝑁𝑁 − 𝑚𝑚 𝑁𝑁 𝑚𝑚
– 𝑝𝑝𝑝𝑝
j
Geometric
Hypergeometric w@ A ⋅ @(1 − 𝑝𝑝) HZ3
Az{𝑝𝑝
@ A 𝑛𝑛 ⋅ 1 – – 𝑝𝑝𝑒𝑒 –
𝑥𝑥 𝑛𝑛 − 𝑥𝑥 𝑛𝑛 𝑁𝑁𝑝𝑝 1 − 𝑝𝑝
1 − (1 − 𝑝𝑝)𝑒𝑒 j 1 − (1 − 𝑝𝑝)𝑡𝑡
1 j 𝑝𝑝𝑝𝑝 𝑝𝑝 Memoryless property
𝑋𝑋:Geometric
trials; 𝑌𝑌: failures HZ3 1 𝑝𝑝c 𝑝𝑝𝑒𝑒 𝑝𝑝
(1 −
(1𝑝𝑝)
𝑋𝑋 = 𝑌𝑌 + 1 − 𝑝𝑝) á𝑝𝑝 𝑝𝑝 𝑝𝑝 − 1 1 − 𝑝𝑝 1− (1(1 −− 𝑝𝑝)𝑒𝑒

j 1− (1(1 −− 𝑝𝑝)𝑡𝑡
𝑝𝑝 1− 𝑝𝑝)𝑒𝑒 j 1− 𝑝𝑝)𝑡𝑡 Memoryless property
𝑋𝑋: trials; 𝑌𝑌: failures 1 𝑟𝑟 𝑝𝑝c 𝑝𝑝𝑝𝑝𝑒𝑒 j ä 𝑝𝑝𝑝𝑝𝑝𝑝 ä
𝑥𝑥 − 1 𝑝𝑝) á â
𝑋𝑋 = 𝑌𝑌 + 1
Negative Binomial à (1 − â 𝑝𝑝ä (1𝑝𝑝 − 𝑝𝑝) HZä − 1
𝑝𝑝 𝑝𝑝 1ã1−−(1(1−−𝑝𝑝)𝑒𝑒 j å 1à1−−(1(1−−𝑝𝑝)𝑡𝑡
𝑟𝑟 − 1 1 − 𝑝𝑝 𝑝𝑝)𝑒𝑒 äj 𝑝𝑝)𝑡𝑡 Neg Bin(𝑟𝑟 = 1, 𝑝𝑝) ~
𝑥𝑥 − 1 ä 𝑟𝑟 𝑟𝑟 à c â 𝑝𝑝𝑒𝑒 j 𝑝𝑝𝑝𝑝 ä
𝑋𝑋: trials; 𝑌𝑌: failures
Negative Binomial à 𝑦𝑦 + 𝑟𝑟â − 𝑝𝑝 1(1 −ä 𝑝𝑝) HZä á 𝑟𝑟 𝑝𝑝 ã 𝑝𝑝 å ä à 𝑝𝑝 â ä Geometric(𝑝𝑝)
𝑋𝑋 = 𝑌𝑌 + 𝑟𝑟 à𝑟𝑟 − 1 â 𝑝𝑝 (1 − 𝑝𝑝) 𝑝𝑝 − 𝑟𝑟 1 − 𝑝𝑝 1à− (1 − 𝑝𝑝)𝑒𝑒 j j â 1à− (1 − 𝑝𝑝)𝑡𝑡 â
𝑟𝑟 − 1 𝑝𝑝 1 − (1 − 𝑝𝑝)𝑒𝑒 1 − (1 − 𝑝𝑝)𝑡𝑡 Neg Bin(𝑟𝑟 = 1, 𝑝𝑝) ~
𝑟𝑟 à c â
𝑋𝑋: trials; 𝑌𝑌: failures 𝑦𝑦 + 𝑟𝑟 − 1 ä 𝑟𝑟 𝑝𝑝 𝑝𝑝 ä 𝑝𝑝 ä Geometric(𝑝𝑝)
𝑋𝑋 = 𝑌𝑌 + 𝑟𝑟 à â 𝑝𝑝 (1 − 𝑝𝑝) á − 𝑟𝑟 à â à â
𝑟𝑟 − 1 𝑒𝑒 Zè ⋅ 𝜆𝜆H 𝑝𝑝 1 − (1 − 𝑝𝑝)𝑒𝑒 j 1 − (1 − 𝑝𝑝)𝑡𝑡 Sum of independent
í
Poisson 𝜆𝜆 𝜆𝜆 𝑒𝑒 èrë Z3s 𝑒𝑒 è(jZ3) Poissons ~
𝑥𝑥! Poisson(𝜆𝜆 = ∑1023 𝜆𝜆0 )
Sum of independent
Zè H
𝑒𝑒 ⋅ 𝜆𝜆 í Z3s
Poisson 𝜆𝜆 𝜆𝜆 𝑒𝑒 èrë 𝑒𝑒 è(jZ3) Poissons ~
𝑥𝑥! Poisson(𝜆𝜆 = ∑1023 𝜆𝜆0 )

www.coachingactuaries.com Copyright © 2018 Coaching Actuaries. All Rights Reserved. 1

Continuous Distributions
Continuous Distributions
Continuous Distributions
PDF
PDF
PDF CDF
CDF
CDF Mean
Mean
Mean Variance
Variance
Variance MGF
MGF
MGF Special Properties
Special Properties
Special Properties

Continuous
Continuous
Continuous 11 1 𝑥𝑥 𝑥𝑥−−
𝑥𝑥𝑎𝑎 −
𝑎𝑎 𝑎𝑎 𝑎𝑎 𝑎𝑎++
𝑎𝑎𝑏𝑏 +
𝑏𝑏 𝑏𝑏 (𝑏𝑏(𝑏𝑏
−(𝑏𝑏
−
𝑎𝑎)−c c c
𝑎𝑎)𝑎𝑎) 𝑒𝑒 mj
𝑒𝑒 mj−
𝑒𝑒 mj
−𝑒𝑒 kj
𝑒𝑒 kj𝑒𝑒 kj
− (𝑋𝑋|𝑋𝑋
(𝑋𝑋|𝑋𝑋
(𝑋𝑋|𝑋𝑋
>>𝑐𝑐) ~ Uniform(𝑐𝑐,
>
𝑐𝑐) ~ Uniform(𝑐𝑐,
𝑐𝑐) ~ Uniform(𝑐𝑐,
𝑏𝑏) 𝑏𝑏) 𝑏𝑏)

Uniform
Uniform
Uniform 𝑏𝑏 𝑏𝑏
−− 𝑏𝑏𝑎𝑎 −
𝑎𝑎 𝑎𝑎 𝑏𝑏 𝑏𝑏−−
𝑏𝑏𝑎𝑎 −
𝑎𝑎 𝑎𝑎 22 2 121212 𝑡𝑡(𝑏𝑏
𝑡𝑡(𝑏𝑏
−
𝑡𝑡(𝑏𝑏
−𝑎𝑎)−
𝑎𝑎)𝑎𝑎) (𝑋𝑋(𝑋𝑋−(𝑋𝑋
−𝑐𝑐|𝑋𝑋
−
𝑐𝑐|𝑋𝑋
𝑐𝑐|𝑋𝑋
>>𝑐𝑐) ~ Uniform(0,
>
𝑐𝑐) ~ Uniform(0,
𝑐𝑐) ~ Uniform(0,
𝑏𝑏 𝑏𝑏
−− 𝑏𝑏𝑐𝑐) −
𝑐𝑐) 𝑐𝑐)
1 1Z1HZHZH H H H 11 1 Memoryless property:
Memoryless property:
Memoryless property:
Exponential
Exponential
Exponential 𝑒𝑒 𝑒𝑒π 𝑒𝑒π π 11
−−1𝑒𝑒 Z−
𝑒𝑒πZ 𝑒𝑒πZ π 𝜃𝜃 𝜃𝜃 𝜃𝜃 𝜃𝜃 c𝜃𝜃 c𝜃𝜃 c
𝜃𝜃 𝜃𝜃 𝜃𝜃 11
−−1𝜃𝜃𝜃𝜃−
𝜃𝜃𝜃𝜃𝜃𝜃𝜃𝜃 (𝑋𝑋(𝑋𝑋−(𝑋𝑋
−𝑎𝑎|𝑋𝑋
−𝑎𝑎|𝑋𝑋
𝑎𝑎|𝑋𝑋
>>𝑎𝑎) ~ 𝑋𝑋
>
𝑎𝑎) ~ 𝑋𝑋
𝑎𝑎) ~ 𝑋𝑋
∫Z3
∫Z3
∫Z3

𝑥𝑥 ∫Z3
𝑥𝑥 ∫Z3
𝑥𝑥 ∫Z3 ZHZHZH 11
−−1/
−/Pr(𝑌𝑌
/Pr(𝑌𝑌
Pr(𝑌𝑌
==𝑘𝑘)=
𝑘𝑘)
, 𝑘𝑘)
, , 1 1 1∫ ∫ ∫ Sum of 𝛼𝛼 independent exponentials(𝜃𝜃) ~
Sum of 𝛼𝛼 independent exponentials(𝜃𝜃) ~
Sum of 𝛼𝛼 independent exponentials(𝜃𝜃) ~
c c c
Gamma
Gamma
Gamma ⋅ 𝑒𝑒⋅ 𝑒𝑒π⋅ 𝑒𝑒π π 𝛼𝛼𝛼𝛼
𝛼𝛼𝛼𝛼
𝛼𝛼𝛼𝛼 𝛼𝛼𝜃𝜃𝛼𝛼𝜃𝜃
𝛼𝛼𝜃𝜃
à à à â â â
Γ(𝛼𝛼)
Γ(𝛼𝛼)⋅ 𝜃𝜃⋅ ∫𝜃𝜃⋅∫𝜃𝜃 ∫
Γ(𝛼𝛼) 82\
82\
82\
H H H 11−−1𝜃𝜃𝜃𝜃−
𝜃𝜃𝜃𝜃𝜃𝜃𝜃𝜃 Gamma(𝛼𝛼,
Gamma(𝛼𝛼,𝜃𝜃) 𝜃𝜃) 𝜃𝜃)
Gamma(𝛼𝛼,
𝑌𝑌 ~ Poisson@𝜆𝜆
𝑌𝑌 ~ Poisson@𝜆𝜆
𝑌𝑌 ~ Poisson@𝜆𝜆
==A =A A
π π π

Symmetry:
Symmetry:
Symmetry:
𝑋𝑋 𝑋𝑋
−− 𝑋𝑋𝜇𝜇 −
𝜇𝜇 𝜇𝜇 Pr(𝑍𝑍
Pr(𝑍𝑍
Pr(𝑍𝑍
≤≤𝑧𝑧)≤
𝑧𝑧)
=𝑧𝑧)
=Pr(𝑍𝑍
=
Pr(𝑍𝑍
Pr(𝑍𝑍
≥≥−𝑧𝑧)
≥
−𝑧𝑧)
−𝑧𝑧)
11 1 (HZæ) ñ ñ ñ
(HZæ)
(HZæ) 𝑍𝑍 𝑍𝑍=𝑍𝑍
== ø ñø
j ññ jøññ j ñ
Normal
Normal
Normal
Z Z Zñ ñ ñ
⋅ 𝑒𝑒⋅ 𝑒𝑒⋅ 𝑒𝑒 𝜎𝜎 𝜎𝜎 𝜎𝜎 𝜇𝜇 𝜇𝜇 𝜇𝜇 𝜎𝜎 c𝜎𝜎 c𝜎𝜎 c Pr(𝑍𝑍
Pr(𝑍𝑍
Pr(𝑍𝑍
≤≤−𝑧𝑧)
≤
−𝑧𝑧)
−𝑧𝑧)
==Pr(𝑍𝑍
=
Pr(𝑍𝑍
Pr(𝑍𝑍
≥≥𝑧𝑧)
≥𝑧𝑧) 𝑧𝑧)
𝜎𝜎√2𝜋𝜋
𝜎𝜎√2𝜋𝜋
𝜎𝜎√2𝜋𝜋
cøcøcø
𝑒𝑒 æjl
𝑒𝑒 æjl
𝑒𝑒 æjl
c c c

Pr(𝑍𝑍
Pr(𝑍𝑍
Pr(𝑍𝑍
≤≤𝑧𝑧)≤
𝑧𝑧)=𝑧𝑧)
=Φ(𝑧𝑧)
=
Φ(𝑧𝑧)
Φ(𝑧𝑧) Sum of independent normals ~
Sum of independent normals ~
Sum of independent normals ~
1
∑1023
∑𝜇𝜇1023 c c∑1∑1023
∑𝜎𝜎1023
c c ) c )
Normal(𝜇𝜇
Normal(𝜇𝜇==∑=
Normal(𝜇𝜇 023 , c𝜎𝜎
0𝜇𝜇, 0𝜎𝜎𝜇𝜇 0 ,=𝜎𝜎= =
023 0𝜎𝜎)
0 𝜎𝜎0

MULTIVARIATE
MULTIVARIATE
MULTIVARIATE PROBABILITY
PROBABILITY
PROBABILITY DISTRIBUTIONS
DISTRIBUTIONS
DISTRIBUTIONS
MULTIVARIATE PROBABILITY DISTRIBUTIONS

*Joint PMF and CDF

*Joint PMF and CDF
*Joint PMF and CDF Independence
Independence
Independence Bivariate Continuous Uniform
Bivariate Continuous Uniform
Bivariate Continuous Uniform
∑EFF H
∑EFF H
∑∑ ∑EFF á
EFF H ∑𝑝𝑝
EFF á (𝑥𝑥,
𝑝𝑝C,n
EFF á
C,n 𝑝𝑝 (𝑥𝑥,
C,n (𝑥𝑥,
𝑦𝑦)𝑦𝑦)=𝑦𝑦)
=1 =1 1 (𝑥𝑥,
𝐹𝐹C,n
𝐹𝐹C,n (𝑥𝑥,
𝐹𝐹C,n (𝑥𝑥,
𝑦𝑦) 𝑦𝑦)=𝑦𝑦) 𝐹𝐹(𝑥𝑥)
=𝐹𝐹C= (𝑥𝑥)
C𝐹𝐹 C ⋅(𝑥𝑥)
𝐹𝐹⋅n𝐹𝐹(𝑦𝑦)
⋅n𝐹𝐹(𝑦𝑦)
n (𝑦𝑦) (𝑥𝑥,
𝑓𝑓C,n
𝑓𝑓C,n (𝑥𝑥,
𝑓𝑓C,n (𝑥𝑥,
𝑦𝑦)𝑦𝑦)=𝑦𝑦)
=1 / Area of domain
=
1 / Area of domain
1 / Area of domain
𝐹𝐹C,n (𝑥𝑥,
𝐹𝐹C,n (𝑥𝑥,
𝐹𝐹C,n (𝑥𝑥,
𝑦𝑦) =∑=
𝑦𝑦)=𝑦𝑦) ∑îOH
îOH ∑∑îOH
∑jOá
jOá ∑𝑝𝑝jOá (𝑠𝑠,
𝑝𝑝C,n
C,n 𝑝𝑝 (𝑠𝑠,
C,n𝑡𝑡)(𝑠𝑠,
𝑡𝑡)
𝑡𝑡) 𝑓𝑓C,n (𝑥𝑥,
𝑓𝑓C,n (𝑥𝑥,
𝑓𝑓C,n (𝑥𝑥,
𝑦𝑦)𝑦𝑦)=𝑦𝑦)
=𝑓𝑓C=𝑓𝑓(𝑥𝑥)
(𝑥𝑥)
C ⋅(𝑥𝑥)
C 𝑓𝑓 𝑓𝑓⋅n𝑓𝑓(𝑦𝑦)
(𝑦𝑦)
⋅n 𝑓𝑓n (𝑦𝑦) Pr(region) = Area of region / Area of domain
Pr(region) = Area of region / Area of domain
Pr(region) = Area of region / Area of domain

ï ñï ñ ï ñ 𝐸𝐸[ℎ(𝑋𝑋)
𝐸𝐸[ℎ(𝑋𝑋)
𝐸𝐸[ℎ(𝑋𝑋) ⋅ 𝑘𝑘(𝑌𝑌)]
⋅ 𝑘𝑘(𝑌𝑌)]
⋅ 𝑘𝑘(𝑌𝑌)] ==𝐸𝐸[ℎ(𝑋𝑋)]
=𝐸𝐸[ℎ(𝑋𝑋)]
𝐸𝐸[ℎ(𝑋𝑋)] ⋅ 𝐸𝐸[𝑘𝑘(𝑌𝑌)]
⋅ 𝐸𝐸[𝑘𝑘(𝑌𝑌)]
⋅ 𝐸𝐸[𝑘𝑘(𝑌𝑌)]
𝐹𝐹C,n (𝑥𝑥,
𝐹𝐹C,n (𝑥𝑥,
𝐹𝐹C,n (𝑥𝑥,
𝑦𝑦)𝑦𝑦)=𝑦𝑦)
=𝑓𝑓C,n
= (𝑥𝑥,
𝑓𝑓C,n (𝑥𝑥,
𝑓𝑓C,n (𝑥𝑥,
𝑦𝑦) (continuous)
𝑦𝑦) (continuous)
𝑦𝑦) (continuous) Bivariate Normal
Bivariate Normal
Bivariate Normal
ïH ïá
ïH ïá
ïH ïá 𝑀𝑀C,n
𝑀𝑀C,n (𝑠𝑠,
𝑀𝑀 (𝑠𝑠,(𝑠𝑠,
𝑡𝑡) 𝑡𝑡)=𝑡𝑡)
=𝑀𝑀= 𝑀𝑀(𝑠𝑠)
𝑀𝑀(𝑠𝑠)
C⋅(𝑠𝑠) ⋅ 𝑀𝑀
𝑀𝑀 n⋅(𝑡𝑡)
n𝑀𝑀(𝑡𝑡)
n (𝑡𝑡)

C,n C C For 𝑋𝑋 ~ Normal(𝜇𝜇C ,C𝜎𝜎,CC
For 𝑋𝑋 ~ Normal(𝜇𝜇
For 𝑋𝑋 ~ Normal(𝜇𝜇 𝜎𝜎cC) and 𝑌𝑌 ~ Normal(𝜇𝜇
c ) and 𝑌𝑌 ~ Normal(𝜇𝜇 c ),
𝜎𝜎cC,) and 𝑌𝑌 ~ Normal(𝜇𝜇 n , n𝜎𝜎,n𝜎𝜎
nn, 𝜎𝜎cn), c ),
*Marginal Distributions and
*Marginal Distributions and
*Marginal Distributions and 𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋,
𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋,
𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋, 𝑌𝑌]𝑌𝑌]=𝑌𝑌]=0,= 0, 0, 𝜌𝜌C,n 𝜌𝜌C,n𝜌𝜌=C,n
=0 =0 0
(𝑌𝑌|𝑋𝑋
(𝑌𝑌|𝑋𝑋
(𝑌𝑌|𝑋𝑋
==𝑥𝑥) ~ Normal, where
=
𝑥𝑥) ~ Normal, where
𝑥𝑥) ~ Normal, where
Conditional Distributions
Conditional Distributions
Conditional Distributions *Joint MGF
*Joint MGF
*Joint MGF 𝑥𝑥 𝑥𝑥
−− 𝑥𝑥𝜇𝜇C−
𝜇𝜇C𝜇𝜇C
𝑝𝑝C𝑝𝑝(𝑥𝑥)
(𝑥𝑥)
C𝑝𝑝 C (𝑥𝑥)
==∑= ∑EFF á
∑𝑝𝑝
EFF á EFF á
C,n (𝑥𝑥,
𝑝𝑝C,n
𝑝𝑝 (𝑥𝑥,
C,n (𝑥𝑥,
𝑦𝑦)
𝑦𝑦) 𝑦𝑦)
𝑀𝑀𝑀𝑀 (𝑠𝑠,
𝑀𝑀 (𝑠𝑠,
𝑡𝑡)(𝑠𝑠,
𝑡𝑡)=𝑡𝑡)
=𝐸𝐸[𝑒𝑒
= îCljn
𝐸𝐸[𝑒𝑒 îCljn
𝐸𝐸[𝑒𝑒 îCljn
] ] ] 𝐸𝐸[𝑌𝑌|𝑋𝑋
𝐸𝐸[𝑌𝑌|𝑋𝑋
𝐸𝐸[𝑌𝑌|𝑋𝑋 ==𝑥𝑥]=𝑥𝑥]=𝑥𝑥]
=𝜇𝜇n=
𝜇𝜇+
n𝜇𝜇+ 𝜌𝜌 +
n 𝜌𝜌
⋅ 𝜎𝜎⋅𝜌𝜌n𝜎𝜎⋅àn𝜎𝜎àn à â â â
C,nC,n
C,n 𝜎𝜎C𝜎𝜎C𝜎𝜎C
𝑝𝑝n𝑝𝑝(𝑦𝑦)
(𝑦𝑦)
n𝑝𝑝 n (𝑦𝑦)
==∑= ∑EFF H
∑𝑝𝑝
EFF H (𝑥𝑥,
𝑝𝑝C,n
EFF H
C,n (𝑥𝑥,
𝑝𝑝C,n (𝑥𝑥,
𝑦𝑦)𝑦𝑦)
𝑦𝑦)
ï ï ï
𝐸𝐸[𝑋𝑋]
𝐸𝐸[𝑋𝑋]
𝐸𝐸[𝑋𝑋] == =𝑀𝑀𝑀𝑀
C,n𝑀𝑀
(𝑠𝑠,
C,n (𝑠𝑠,
C,n 𝑡𝑡)ù
(𝑠𝑠,
𝑡𝑡)ù𝑡𝑡)ù 𝑉𝑉𝑉𝑉𝑉𝑉[𝑌𝑌|𝑋𝑋
𝑉𝑉𝑉𝑉𝑉𝑉[𝑌𝑌|𝑋𝑋
𝑉𝑉𝑉𝑉𝑉𝑉[𝑌𝑌|𝑋𝑋 ==𝑥𝑥]= 𝑥𝑥]=𝑥𝑥]
=𝜎𝜎n= 𝜎𝜎cn(1
𝜎𝜎cn(1 c (1 c ) c ) c )
− − 𝜌𝜌 −𝜌𝜌 𝜌𝜌
𝑝𝑝(𝑥𝑥|𝑌𝑌
𝑝𝑝C|n
𝑝𝑝C|n (𝑥𝑥|𝑌𝑌
C|n (𝑥𝑥|𝑌𝑌
==𝑦𝑦)= 𝑦𝑦)=𝑦𝑦)
=𝑝𝑝C,n
= (𝑥𝑥,
𝑝𝑝C,n (𝑥𝑥,
𝑝𝑝C,n (𝑥𝑥,
𝑦𝑦) ⁄𝑝𝑝𝑦𝑦)
𝑦𝑦) ⁄n𝑝𝑝(𝑦𝑦)
⁄n𝑝𝑝(𝑦𝑦)
n (𝑦𝑦)
ïîïî ïî î2j2\
î2j2\
î2j2\

𝐸𝐸[𝑌𝑌]
𝐸𝐸[𝑌𝑌]
𝐸𝐸[𝑌𝑌]
ï ï ï
== =𝑀𝑀𝑀𝑀 𝑀𝑀
(𝑠𝑠,
(𝑠𝑠,
𝑡𝑡)ù
(𝑠𝑠,
𝑡𝑡)ù𝑡𝑡)ù Expectation and Variance for Sum and Average
Expectation and Variance for Sum and Average
Expectation and Variance for Sum and Average
*Joint Expected Value and
*Joint Expected Value and
*Joint Expected Value and ïj ïj ïjC,nC,nC,n
î2j2\
î2j2\
î2j2\ of I.I.D. Random Variables
of I.I.D. Random Variables
of I.I.D. Random Variables
Conditional Expectation
Conditional Expectation
Conditional Expectation ï ü†°
ï ü†°
ï ü†°
𝐸𝐸[𝑋𝑋 û û1û
𝐸𝐸[𝑋𝑋𝑌𝑌 𝑌𝑌]1𝑌𝑌=
𝐸𝐸[𝑋𝑋 ] 1=] = 𝑀𝑀𝑀𝑀 (𝑠𝑠,(𝑠𝑠,
𝑀𝑀 (𝑠𝑠,
𝑡𝑡)ù𝑡𝑡)ù𝑡𝑡)ù 𝑆𝑆 𝑆𝑆==
𝑆𝑆𝑋𝑋3=
𝑋𝑋+
3𝑋𝑋+
⋯
3+⋯
+⋯+
𝑋𝑋1+
𝑋𝑋 1𝑋𝑋 1 𝑋𝑋≤ ≤==
𝑋𝑋 ≤[𝑋𝑋=
𝑋𝑋 [𝑋𝑋
3+ [𝑋𝑋
3+ ⋯
3+
⋯+⋯
+
𝑋𝑋1+ 1 𝑛𝑛
𝑋𝑋]1⁄𝑋𝑋]𝑛𝑛⁄]⁄ 𝑛𝑛
YY Y Y Y ïîü ü
ïîïj °ü°
ïîïj ïjC,n
° C,nC,n
𝐸𝐸[𝑔𝑔(𝑋𝑋,
𝐸𝐸[𝑔𝑔(𝑋𝑋,
𝐸𝐸[𝑔𝑔(𝑋𝑋, 𝑌𝑌)]
𝑌𝑌)] 𝑌𝑌)]
==∫ZY =∫ZY
∫ZY∫ZY ∫ 𝑔𝑔(𝑥𝑥,
𝑔𝑔(𝑥𝑥, 𝑔𝑔(𝑥𝑥,
𝑦𝑦)𝑦𝑦)
⋅ 𝑦𝑦)
𝑓𝑓⋅C,n⋅ (𝑥𝑥,
𝑓𝑓C,n (𝑥𝑥,
𝑓𝑓C,n (𝑥𝑥,
𝑦𝑦) 𝑑𝑑𝑑𝑑
𝑦𝑦) 𝑑𝑑𝑑𝑑
𝑦𝑦) 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
î2j2\
î2j2\
î2j2\
ZY 𝑀𝑀𝑀𝑀 (𝑡𝑡,
𝑀𝑀 (𝑡𝑡,
𝑡𝑡)(𝑡𝑡,
𝑡𝑡)=𝑡𝑡)
=𝑀𝑀=
𝑀𝑀 𝑀𝑀(𝑡𝑡)
(𝑡𝑡)
(𝑡𝑡) 𝐸𝐸[𝑆𝑆]
𝐸𝐸[𝑆𝑆]
𝐸𝐸[𝑆𝑆]
==𝑛𝑛 =𝑛𝑛
⋅ 𝐸𝐸[𝑋𝑋
⋅𝑛𝑛𝐸𝐸[𝑋𝑋
⋅ 0𝐸𝐸[𝑋𝑋
] 0 ] 0 ] 𝐸𝐸[𝑋𝑋 ≤]≤=
𝐸𝐸[𝑋𝑋 ] ≤=]𝐸𝐸[𝑋𝑋
𝐸𝐸[𝑋𝑋 =𝐸𝐸[𝑋𝑋] 0] 0]
YY Y C,nC,n
C,n Cln
ClnCln 0𝐸𝐸[𝑋𝑋
𝐸𝐸[𝑋𝑋|𝑌𝑌 ==𝑦𝑦]=
𝐸𝐸[𝑋𝑋|𝑌𝑌
𝐸𝐸[𝑋𝑋|𝑌𝑌 𝑦𝑦]=𝑦𝑦]
=∫ZY= 𝑥𝑥ZY
∫ZY
∫ 𝑥𝑥⋅ 𝑓𝑓⋅𝑥𝑥 (𝑥𝑥|𝑌𝑌
𝑓𝑓C|n (𝑥𝑥|𝑌𝑌
(𝑥𝑥|𝑌𝑌
⋅ 𝑓𝑓C|n ==𝑦𝑦) 𝑑𝑑𝑑𝑑 =𝑦𝑦) 𝑑𝑑𝑑𝑑
𝑦𝑦) 𝑑𝑑𝑑𝑑

≤]≤=

C|n
Multivariate Transformation
Multivariate Transformation
Multivariate Transformation 𝑉𝑉𝑉𝑉𝑉𝑉[𝑆𝑆]
𝑉𝑉𝑉𝑉𝑉𝑉[𝑆𝑆]
𝑉𝑉𝑉𝑉𝑉𝑉[𝑆𝑆]
==𝑛𝑛 = 𝑛𝑛
⋅ 𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋 0 ] 𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋
⋅𝑛𝑛𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋
⋅ 𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋0 ] 𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋 ] ≤=](1/𝑛𝑛)
0 ] 𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋 =
(1/𝑛𝑛)
(1/𝑛𝑛) 0 ] 0 ] 0 ]
⋅ 𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋
⋅ 𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋
⋅ 𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋

Double Expectation and
Double Expectation and
Double Expectation and 𝑓𝑓¢𝑓𝑓£¢𝑓𝑓£¢
,¢ ,¢(𝑤𝑤(𝑤𝑤,(𝑤𝑤
𝑤𝑤, 𝑤𝑤), 𝑤𝑤
=)= )𝑓𝑓 = C[ℎ
𝑓𝑓,C£𝑓𝑓ñ,C [ℎ
3ñ(𝑤𝑤[ℎ(𝑤𝑤
33,(𝑤𝑤
3𝑤𝑤 3),
, c𝑤𝑤,c𝑤𝑤),
ℎccℎ),(𝑤𝑤
cℎ(𝑤𝑤
3c,(𝑤𝑤
3𝑤𝑤 3)]
,c𝑤𝑤)]
, c𝑤𝑤 ⋅c| 𝐽𝐽|
)]
⋅ | 𝐽𝐽|
⋅ | 𝐽𝐽| Central Limit Theorem
Central Limit Theorem
Central Limit Theorem
ñ £ñ,¢3ñ 3 c3 c c C£C £ñ,C 3
Law of Total Variance
Law of Total Variance
Law of Total Variance where 𝑥𝑥
where 𝑥𝑥
where 𝑥𝑥 3 =3 = 3ℎ3= ℎ(𝑤𝑤
3ℎ(𝑤𝑤33,(𝑤𝑤
3𝑤𝑤 3),
, c𝑤𝑤 ,c𝑤𝑤),
𝑥𝑥cc𝑥𝑥),=
c𝑥𝑥=cℎcℎ(𝑤𝑤
= cℎ(𝑤𝑤
3c,(𝑤𝑤 3),
,c𝑤𝑤), c ),
3𝑤𝑤, c𝑤𝑤 The sum of a large number of identically and
The sum of a large number of identically and
The sum of a large number of identically and
𝐸𝐸[𝑋𝑋]
𝐸𝐸[𝑋𝑋]==𝐸𝐸ó𝐸𝐸[𝑋𝑋|𝑌𝑌]ò
𝐸𝐸[𝑋𝑋] =
𝐸𝐸ó𝐸𝐸[𝑋𝑋|𝑌𝑌]ò
𝐸𝐸ó𝐸𝐸[𝑋𝑋|𝑌𝑌]ò ïHïH
£ £ ïHïH
£ ïH£ £ ïH£
independently distributed (i.i.d.) random variables
independently distributed (i.i.d.) random variables
independently distributed (i.i.d.) random variables
ïßïß £ ïß
£ ïß
£ ïßñ ïß
𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋]
𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋]==𝐸𝐸ó𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋|𝑌𝑌]ò
𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋] = ++
𝐸𝐸ó𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋|𝑌𝑌]ò 𝑉𝑉𝑉𝑉𝑉𝑉ó𝐸𝐸[𝑋𝑋|𝑌𝑌]ò
𝐸𝐸ó𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋|𝑌𝑌]ò +
𝑉𝑉𝑉𝑉𝑉𝑉ó𝐸𝐸[𝑋𝑋|𝑌𝑌]ò
𝑉𝑉𝑉𝑉𝑉𝑉ó𝐸𝐸[𝑋𝑋|𝑌𝑌]ò 𝐽𝐽 =
𝐽𝐽 =𝐽𝐽¶ ïH
=¶ ïH¶ ïH ¶ ñ ¶ ñ ¶ approximately follows a normal distribution
approximately follows a normal distribution
approximately follows a normal distribution
ïHïHïHñ ñ ñ ñ ñ ñ

Covariance and Correlation Coefficient

Covariance and Correlation Coefficient
Covariance and Correlation Coefficient
ïßïß
£ ïß
£ ïß
£ ïß
ñ ïß
ñ ñ Order Statistics
Order Statistics
Order Statistics
𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋,
𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋,
𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋, 𝑌𝑌]𝑌𝑌]=𝑌𝑌]
=𝐸𝐸[𝑋𝑋𝑋𝑋]
=
𝐸𝐸[𝑋𝑋𝑋𝑋]
𝐸𝐸[𝑋𝑋𝑋𝑋]
−− −
𝐸𝐸[𝑋𝑋]𝐸𝐸[𝑌𝑌]
𝐸𝐸[𝑋𝑋]𝐸𝐸[𝑌𝑌]
𝐸𝐸[𝑋𝑋]𝐸𝐸[𝑌𝑌] Multinomial Distribution
Multinomial Distribution
Multinomial Distribution 𝑋𝑋(3)
𝑋𝑋(3) 𝑋𝑋=(3)
=min(𝑋𝑋
=
min(𝑋𝑋
min(𝑋𝑋
3 , 3𝑋𝑋,c𝑋𝑋
3,,c…𝑋𝑋, … ,1𝑋𝑋) ,1𝑋𝑋) 1 )
c, ,𝑋𝑋…
𝐶𝐶𝐶𝐶𝐶𝐶[𝑎𝑎𝑎𝑎,
𝐶𝐶𝐶𝐶𝐶𝐶[𝑎𝑎𝑎𝑎,
𝐶𝐶𝐶𝐶𝐶𝐶[𝑎𝑎𝑎𝑎, 𝑏𝑏𝑏𝑏]
𝑏𝑏𝑏𝑏] 𝑏𝑏𝑏𝑏]
==𝑎𝑎𝑎𝑎
=𝑎𝑎𝑎𝑎
⋅𝑎𝑎𝑎𝑎
⋅ 𝐶𝐶𝐶𝐶𝑣𝑣[𝑋𝑋,
⋅ 𝐶𝐶𝐶𝐶𝑣𝑣[𝑋𝑋,
𝐶𝐶𝐶𝐶𝑣𝑣[𝑋𝑋, 𝑌𝑌] 𝑌𝑌] 𝑌𝑌] Pr(𝑋𝑋
Pr(𝑋𝑋 Pr(𝑋𝑋 𝑥𝑥) 8𝑥𝑥) 8 ) 𝑋𝑋(1)
𝑋𝑋(1) 𝑋𝑋=(1)
=max(𝑋𝑋
=
max(𝑋𝑋
max(𝑋𝑋
3, 3𝑋𝑋, c𝑋𝑋
3,,c…𝑋𝑋, … , 1𝑋𝑋) ,1𝑋𝑋) 1 )
c, ,𝑋𝑋…
3 = 3 = 3𝑥𝑥3= 𝑥𝑥, 3…𝑥𝑥, …3, ,𝑋𝑋…
, 8𝑋𝑋,8=𝑋𝑋= 8𝑥𝑥8 =
𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋,
𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋,
𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋, 𝑋𝑋]𝑋𝑋]=𝑋𝑋]
=𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋]
=
𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋]
𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋] 𝑛𝑛!𝑛𝑛!𝑛𝑛!

𝑉𝑉𝑉𝑉𝑉𝑉[𝑎𝑎𝑎𝑎
𝑉𝑉𝑉𝑉𝑉𝑉[𝑎𝑎𝑎𝑎
𝑉𝑉𝑉𝑉𝑉𝑉[𝑎𝑎𝑎𝑎 ++ + ==𝑎𝑎c=
𝑏𝑏𝑏𝑏]
𝑏𝑏𝑏𝑏] 𝑏𝑏𝑏𝑏] c c
𝑎𝑎𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋]
𝑎𝑎𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋]
𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋]
++ 𝑏𝑏 c+ c c
𝑏𝑏𝑉𝑉𝑉𝑉𝑉𝑉[𝑌𝑌]
𝑏𝑏𝑉𝑉𝑉𝑉𝑉𝑉[𝑌𝑌]
𝑉𝑉𝑉𝑉𝑉𝑉[𝑌𝑌]
++ 2𝑎𝑎𝑎𝑎
+
2𝑎𝑎𝑎𝑎
2𝑎𝑎𝑎𝑎 == = H£H£ H£
⋅ 𝑝𝑝⋅ 𝑝𝑝⋅ 𝑝𝑝⋅ … ⋅… ⋅8𝑝𝑝H⋅8®𝑝𝑝H 8® H®
⋅⋅ 𝑝𝑝… For i.i.d. random variables,
For i.i.d. random variables,
For i.i.d. random variables,
𝑥𝑥3𝑥𝑥! 3⋅𝑥𝑥!…3⋅ !…⋅⋅ 𝑥𝑥…⋅8𝑥𝑥!⋅8𝑥𝑥! 83! 3 3 (𝑥𝑥) 1 1 1
⋅ 𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋,
⋅ 𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋,
⋅ 𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋, 𝑌𝑌] 𝑌𝑌] 𝑌𝑌] 𝑆𝑆C𝑆𝑆(£)
C𝑆𝑆 C(𝑥𝑥)
(£) (£)
(𝑥𝑥)
==[𝑆𝑆=[𝑆𝑆
C (𝑥𝑥)]
[𝑆𝑆(𝑥𝑥)]
C C (𝑥𝑥)]

𝐸𝐸[𝑋𝑋
𝐸𝐸[𝑋𝑋] 0=
0𝐸𝐸[𝑋𝑋 ] 0=]𝑛𝑛𝑝𝑝 = 𝑛𝑛𝑝𝑝
0 𝑛𝑛𝑝𝑝
0 0
CC C
(𝑥𝑥)
(𝑥𝑥)(𝑥𝑥)[𝐹𝐹
[𝐹𝐹
(𝑥𝑥)]
[𝐹𝐹
(𝑥𝑥)]1 1 1
(𝑥𝑥)]

𝐹𝐹C𝐹𝐹(°)𝐹𝐹
C(°)C(°) = = =
𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋,
𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋,
𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋,
𝑌𝑌]𝑌𝑌]𝑌𝑌] 𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋 0 ] 0=
𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋
𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋 ] 0=]𝑛𝑛𝑝𝑝 =
𝑛𝑛𝑝𝑝0 (1 0 (1
𝑛𝑛𝑝𝑝 −0 (1−𝑝𝑝0− 𝑝𝑝) 0 )
𝑝𝑝0 )
𝜌𝜌C,n
𝜌𝜌C,n
𝜌𝜌=C,n
=𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋,
=𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋,
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶[𝑋𝑋,
𝑌𝑌]𝑌𝑌]=𝑌𝑌]
== 𝐶𝐶𝐶𝐶𝐶𝐶ó𝑋𝑋
𝐶𝐶𝐶𝐶𝐶𝐶ó𝑋𝑋
𝐶𝐶𝐶𝐶𝐶𝐶ó𝑋𝑋 ò0 _, =
ò _= ò−𝑛𝑛𝑝𝑝
e𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋]e𝑉𝑉𝑉𝑉𝑉𝑉[𝑌𝑌]
e𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋]e𝑉𝑉𝑉𝑉𝑉𝑉[𝑌𝑌]
e𝑉𝑉𝑉𝑉𝑉𝑉[𝑋𝑋]e𝑉𝑉𝑉𝑉𝑉𝑉[𝑌𝑌] 0 , 𝑋𝑋0 ,_𝑋𝑋 𝑋𝑋 =−𝑛𝑛𝑝𝑝−𝑛𝑛𝑝𝑝
0 𝑝𝑝0_𝑝𝑝
, _0 𝑝𝑝
, _ , 𝑖𝑖 ≠ 𝑖𝑖 ≠𝑖𝑖𝑗𝑗 ≠𝑗𝑗 𝑗𝑗

Insurance
Insurance and
Insurance and
Risk
and Risk
Risk Management
Management
Management INSURANCE AND RISK MANAGEMENT

Category
Category
Category Definition of Payment, 𝒀𝒀
Definition of Payment, 𝒀𝒀
Definition of Payment, 𝒀𝒀 𝑬𝑬[𝒀𝒀]
𝑬𝑬[𝒀𝒀]
𝑬𝑬[𝒀𝒀]
YY Y
0,
0,
0, 𝑋𝑋 𝑋𝑋≤𝑋𝑋
≤𝑑𝑑 ≤
𝑑𝑑 𝑑𝑑 ∫S∫S(𝑥𝑥
∫(𝑥𝑥
S
−(𝑥𝑥
−𝑑𝑑)− 𝑑𝑑) 𝑓𝑓⋅C𝑓𝑓(𝑥𝑥) 𝑑𝑑𝑑𝑑
⋅ 𝑑𝑑) C⋅ 𝑓𝑓(𝑥𝑥) 𝑑𝑑𝑑𝑑
C (𝑥𝑥) 𝑑𝑑𝑑𝑑
For exponential:
For exponential:
For exponential:
Deductible
Deductible
Deductible =ƒ =
𝑌𝑌 𝑌𝑌=𝑌𝑌 ƒ ƒ YY Y
𝑋𝑋 𝑋𝑋−𝑋𝑋−𝑑𝑑,−
𝑑𝑑, 𝑑𝑑, 𝑋𝑋 𝑋𝑋>𝑋𝑋
>𝑑𝑑 >
𝑑𝑑 𝑑𝑑 ∫S∫S𝑆𝑆∫CS𝑆𝑆(𝑥𝑥) 𝑑𝑑𝑑𝑑
C𝑆𝑆(𝑥𝑥) 𝑑𝑑𝑑𝑑
C (𝑥𝑥) 𝑑𝑑𝑑𝑑
𝜃𝜃 𝜃𝜃
⋅ Pr(𝑋𝑋
⋅𝜃𝜃Pr(𝑋𝑋
⋅ Pr(𝑋𝑋
>>𝑑𝑑)
>
𝑑𝑑) 𝑑𝑑)
« « «
𝑋𝑋,𝑋𝑋,𝑋𝑋, 𝑋𝑋 𝑋𝑋<𝑋𝑋<𝑢𝑢 <
𝑢𝑢 𝑢𝑢 ∫\∫\𝑥𝑥∫𝑥𝑥\⋅ 𝑓𝑓⋅𝑥𝑥
C𝑓𝑓(𝑥𝑥) 𝑑𝑑𝑑𝑑
C⋅ 𝑓𝑓(𝑥𝑥) 𝑑𝑑𝑑𝑑
C (𝑥𝑥) 𝑑𝑑𝑑𝑑
++ 𝑢𝑢 +⋅ 𝑆𝑆⋅𝑢𝑢C𝑆𝑆(𝑢𝑢)
𝑢𝑢 ⋅C𝑆𝑆(𝑢𝑢)
C (𝑢𝑢) For exponential:
For exponential:
For exponential:
Policy Limit
Policy Limit
Policy Limit =≈ =
𝑌𝑌 𝑌𝑌=𝑌𝑌 ≈ ≈ « « «
𝑢𝑢, 𝑢𝑢, 𝑢𝑢, 𝑋𝑋 𝑋𝑋≥𝑋𝑋
≥𝑢𝑢 ≥
𝑢𝑢 𝑢𝑢 ∫\∫\𝑆𝑆∫ (𝑥𝑥) 𝑑𝑑𝑑𝑑
C𝑆𝑆C (𝑥𝑥) 𝑑𝑑𝑑𝑑
𝑆𝑆C (𝑥𝑥) 𝑑𝑑𝑑𝑑
𝜃𝜃 𝜃𝜃
⋅ Pr(𝑋𝑋
⋅𝜃𝜃Pr(𝑋𝑋
⋅ Pr(𝑋𝑋
<<𝑢𝑢)
<
𝑢𝑢) 𝑢𝑢)
\

Deductible and Policy Limit

Deductible and Policy Limit
Deductible and Policy Limit 0,0, 0, 𝑋𝑋 𝑋𝑋≤𝑋𝑋 ≤𝑑𝑑
≤𝑑𝑑
𝑑𝑑 Sl«
Sl«
Sl«
∫S∫S ∫S(𝑥𝑥(𝑥𝑥
−(𝑥𝑥 𝑓𝑓⋅C𝑓𝑓(𝑥𝑥) 𝑑𝑑𝑑𝑑
(𝑥𝑥) 𝑑𝑑𝑑𝑑
−
𝑑𝑑)−
𝑑𝑑)
⋅ 𝑑𝑑) C C (𝑥𝑥) 𝑑𝑑𝑑𝑑
⋅ 𝑓𝑓 ++ 𝑢𝑢⋅ 𝑆𝑆⋅𝑢𝑢C𝑆𝑆(𝑑𝑑
𝑢𝑢 + ⋅C𝑆𝑆(𝑑𝑑
C+(𝑑𝑑
+𝑢𝑢)
+
𝑢𝑢)
𝑢𝑢) For exponential:
For exponential:
For exponential:
(𝑢𝑢 is the policy limit/
(𝑢𝑢 is the policy limit/
(𝑢𝑢 is the policy limit/ =»𝑋𝑋=
𝑌𝑌 𝑌𝑌=𝑌𝑌 »𝑋𝑋−»𝑋𝑋
−𝑑𝑑,−
𝑑𝑑, 𝑑𝑑, 𝑑𝑑 𝑑𝑑<𝑑𝑑
<𝑋𝑋<𝑋𝑋<𝑋𝑋<𝑑𝑑 <𝑑𝑑++𝑑𝑑𝑢𝑢 +
𝑢𝑢 𝑢𝑢 Sl«
Sl«
Sl«
∫S ∫S ∫S𝑆𝑆C𝑆𝑆(𝑥𝑥) 𝑑𝑑𝑑𝑑
(𝑥𝑥) 𝑑𝑑𝑑𝑑
C𝑆𝑆C (𝑥𝑥) 𝑑𝑑𝑑𝑑
𝜃𝜃 𝜃𝜃
⋅ Pr(𝑑𝑑
⋅𝜃𝜃Pr(𝑑𝑑
⋅ Pr(𝑑𝑑
<<𝑋𝑋<𝑋𝑋<𝑋𝑋
<𝑑𝑑 <
𝑑𝑑++
𝑑𝑑𝑢𝑢)
+
𝑢𝑢) 𝑢𝑢)
maximum payment)
maximum payment)
maximum payment) 𝑢𝑢,𝑢𝑢, 𝑢𝑢, 𝑋𝑋 𝑋𝑋≥𝑋𝑋
≥𝑑𝑑 ≥
𝑑𝑑++𝑑𝑑𝑢𝑢 +
𝑢𝑢 𝑢𝑢

Unreimbursed Loss, 𝒁𝒁
Unreimbursed Loss, 𝒁𝒁
Unreimbursed Loss, 𝒁𝒁
If 𝑋𝑋 is the loss and 𝑌𝑌 is the payment (i.e. reimbursed loss), then 𝑋𝑋 ==𝑌𝑌=
If 𝑋𝑋 is the loss and 𝑌𝑌 is the payment (i.e. reimbursed loss), then 𝑋𝑋
If 𝑋𝑋 is the loss and 𝑌𝑌 is the payment (i.e. reimbursed loss), then 𝑋𝑋 ++
𝑌𝑌 𝑌𝑌𝑍𝑍 +
𝑍𝑍⇒𝑍𝑍
⇒𝑍𝑍⇒
𝑍𝑍=𝑍𝑍
=𝑋𝑋=
𝑋𝑋−𝑋𝑋
−𝑌𝑌, and 𝐸𝐸[𝑍𝑍]
− 𝑌𝑌, and 𝐸𝐸[𝑍𝑍]
==𝐸𝐸[𝑋𝑋]
𝑌𝑌, and 𝐸𝐸[𝑍𝑍] = 𝐸𝐸[𝑋𝑋]
−−
𝐸𝐸[𝑋𝑋] 𝐸𝐸[𝑌𝑌].
− 𝐸𝐸[𝑌𝑌].
𝐸𝐸[𝑌𝑌].

* Learn both the discrete and continuous cases
* Learn both the discrete and continuous cases
* Learn both the discrete and continuous cases

Copyright © 2018 Coaching Actuaries. All Rights Reserved. 2

www.coachingactuaries.com
www.coachingactuaries.com Personal copies
Copyright permitted.
© 2018 Resale
Coaching or distribution
Actuaries. is prohibited.
All Rights Reserved. 2
www.coachingactuaries.com
www.facebook.com/coachingactuaries
blog.coachingactuaries.com
P  Formula  Sheet

1
Copyright  ©  2014  Coaching  Actuaries.    All  Rights  Reserved.
 2
www.coachingactuaries.com Copyright  ©  2014  Coaching  Actuaries.    All  Rights  Reserved.
 3

www.coachingactuaries.com Copyright  ©  2014  Coaching  Actuaries.    All  Rights  Reserved.

 Exam 1/P Formula Sheets
WWW.P ROBABILITY E XAM . COM

S ETS I NTEGRALS C ONT. C OUNTING T ECHNIQUES

De Morgan’s Law Substitution Multiplication Rule
c c c c c c Z b g(b)
(A ∪ B) = A ∩ B (A ∩ B) = A ∪ B
Z
0 A compound experiment consisting of 2 sub-experiments with a
f (g(x))g (x) dx = f (u) du
Inclusion-Exclusion Principle a g(a) and b possible outcomes resp. has a · b possible outcomes.
Integration by Parts Permutations
|A ∪ B| = |A| + |B| − |A ∩ B|
Z b Z b
|A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |B ∩ C| − |A ∩ C| + |A ∩ B ∩ C| b An ordered arrangement of k elements from an n element set
u dv = uv a − v du
a a n!
Count: n Pk = n(n − 1) · · · (n − (k − 1)) =
D ERIVATIVES Other Useful Identities (n − k)!
Z b  cx
xe e cx b
 Combinations
Function Derivative cx
xe dx = − 2  c 6= 0
a c c a A k-element subset of an n element set
c 0 Z ∞  
n! n n!
xr rxr−1 n −cx
x e dx = n+1 n ∈ N, c > 0 Count: n Ck = =
c k k!(n − k)!
cf (x) cf 0 (x) 0

f (x) + g(x) f 0 (x) + g 0 (x) Properties of Binomial Coefficients

f (x) · g(x) f 0 (x) · g(x) + f (x) · g 0 (x) P ROBABILITY A XIOMS n
X n   
n
 
n

n
f (g(x)) f 0 (g(x)) · g 0 (x) =2 =
k k n−k
e x
e x Probability Function Definition k=0
         
1 n−1 n n n−1 n−1
ln x x 1. P (S) = 1 n =k = +
k−1 k k k−1 k
ax x
a · ln a
2. P (A) ≥ 0 for all A 
m+n
 X r  
m n

=
3. For mutually disjoint events A1 , A2 , . . . r k r−k
k=0
I NTEGRALS
∞ ∞
!
[ X Counting Rules
Properties of Integrals P Ai = P (Ai )
Z b i=1 i=1 # of ways to select k elements from n total elements:
c dx = c · (b − a)
a Axiom Consequences
Z b Z b Z b order matters order doesn’t matter
[f (x) + g(x)] dx = f (x) dx + g(x) dx P (∅) = 0  
a a a n+k−1
P (Ac ) = 1 − P (A) replace n k
Z b Z b k
cf (x) dx = c f (x) dx c
A ⊆ B =⇒ P (B ∩ A ) = P (B) − P (A)  
a a n! n
P (A ∪ B) = P (A) + P (B) − P (A ∩ B) don’t replace
Z b Z c Z b (n − k)! k
f (x) dx = f (x) dx + f (x) dx
a a c
P (A ∪ B ∪ C) = P (A) + P (B) + P (C)
− P (A ∩ B) − P (B ∩ C) − P (A ∩ C)
Applications of the FTC
+ P (A ∩ B ∩ C)
r+1 b
Z b  Z b b
r x  (r 6= −1) 1  |A|
x dx = dx = ln |x| If S is finite with equally likely outcomes then P (A) =
a r + 1 
a a x a |S|
Z b b Z b x b

x x
 x c 
e dx = e  c dx =
a a a ln c 
a
 Exam 1/P Formula Sheets
WWW.P ROBABILITY E XAM . COM

C ONDITIONAL P ROBABILITY R ANDOM VARIABLES C ONT. S UMMARY S TATISTICS C ONT.

Definition Discrete Random Variable Variance
P (A ∩ B) X has finite or countably infinite (listable) support. Var(X) = E[(X − µ)2 ] = E[X 2 ] − E[X]2
P (A | B) = provided P (B) > 0
P (B) Probability Mass Function: p(x) = P (X = x) Variance Properties
Conditional Probabilities are Probabilities X
A valid PMF has p(x) ≥ 0 and p(x) = 1 Var(X) ≥ 0 Var(X) = 0 ⇐⇒ P (X = µ) = 1
P (Ac | B) = 1 − P (A | B) x
2
Var(X + c) = Var(X) Var(cX) = c · Var(X)
P (A ∪ B | C) = P (A | C) + P (B | C) − P (A ∩ B | C) Jumps in the CDF are the probabilities (values of the PMF). X, Y independent =⇒ Var(X + Y ) = Var(X) + Var(Y )
Multiplication Rule P (A) > 0, P (B) > 0 Continuous Random Variable
Standard Deviation Coefficient of Variation
P (A ∩ B) = P (A | B) · P (B) = P (B | A) · P (A) X has continuous CDF differentiable except at finitely many points. p σX
σX = Var(X) CV(X) =
Probability Density Function: f (x) = F 0 (x) µX
Bayes’ Theorem
Z ∞
Skew
P (B | A) · P (A) A valid PDF has f (x) ≥ 0 and f (x) = 1
P (A | B) = provided P (A) > 0, P (B) > 0 −∞
" 3 #
P (B) X −µ E[X 3 ] − 3µσ 2 − µ3
Skew(X) = E =
Z b σ σ3
Law of Total Probability P (a ≤ X ≤ b) = f (x) dx = F (b) − F (a)
a
If A1 , A2 , . . . , An partition S with P (Ai ) > 0, then Jensen’s Inequality
Mixed Type Random Variable
P (B) = P (B | A1 ) · P (A1 ) + · · · + P (B | An ) · P (An ) g 00 (x) ≥ 0 =⇒ E[g(X)] ≥ g(E[X])
X’s CDF is a weighted average of a continuous and discrete CDF: 00
Independent Events g (x) ≤ 0 =⇒ E[g(X)] ≤ g(E[X])
F (x) = α · FC (x) + (1 − α) · FD (x) 0<α<1
Definition: P (A ∩ B) = P (A) · P (B) P (g(X) = a + bX) = 1 ⇐⇒ E[g(X)] = g(E[X])

(A, B) indep. pair =⇒ (A, B c ), (Ac , B), (Ac , B c ) also indep. pairs Mode
S UMMARY S TATISTICS A mode is an x value which maximizes the PMF/PDF.
Expected Value It is possible to have 0, 1, 2, . . . or infinite modes.
R ANDOM VARIABLES Z ∞
Discrete: any values with the largest probability
X
A random variable, X, is a function from the sample space S to R E[X] = x · p(x) E[X] = x · f (x) dx
−∞
Cumulative Distribution Function
x Continuous: check end points of interval and where f 0 (x) = 0
Law of the Unconscious Statistician (LOTUS)
F (x) = P (X ≤ x) Percentile
X Z ∞
E[g(X)] = g(x) · p(x) E[g(X)] = g(x) · f (x) dx c is a (100p)th percentile of X if P (X ≤ c) ≥ p and P (X ≥ c) ≥ 1−p
x −∞
A 50th percentile is called a median
Expected Value Linearity
Discrete: look for smallest c with F (c) ≥ p
E[aX + b] = a · E[X] + b E[X + Y ] = E[X] + E[Y ] Continuous: solve for c in F (c) = p
Survival Shortcut
∞
X
If X is nonnegative integer-valued, then E[X] = P (X > k)
k=0
A valid CDF is nondecreasing, right-continuous and Z ∞
If X is nonnegative continuous, then E[X] = [1 − F (x)] dx
lim F (x) = 0, lim F (x) = 1 0
x→−∞ x→∞
 Exam 1/P Formula Sheets
WWW.P ROBABILITY E XAM . COM

C OMMON D ISCRETE D ISTRIBUTIONS

Distribution Description P (X = x) Expected Value Variance MGF Properties

2 at (b+1)t
1 a+b (b − a + 1) − 1 e −e
DUniform({a, . . . , b}) Equally likely values a, . . . , b
b−a+1 2 12 (b − a + 1)(1 − et )

P (X = 1) = p t
Bernoulli(p) 1 trial w/ success chance p p p(1 − p) 1 − p + pe
P (X = 0) = 1 − p
 
# of successes in n indep. n x n−x t n
Binomial(n, p) p (1 − p) np np(1 − p) (1 − p + pe ) np ∈ N =⇒ np = mode = median
Bernoulli(p) trials x
# w/ property chosen w/ K N −K


  K
x n−x K K K N −n Resembles Binomial(n, N)
HyperGeom(N, K, n) out replacement from N N
n n 1− ugly
N −1 with large N relative to n


where K have property n
N N N

e−λ λx λ(et −1) Approximates Binomial(n, p)

Poisson(λ) Common frequency dist. λ λ e
x! when λ = np, n large, p small

# of failures before x 1−p 1−p p

Geometric(p) (1 − p) p Only memoryless discrete dist.
first probability p success p p2 1 − (1 − p)et
   r
# of failures before x+r−1 r x r(1 − p) r(1 − p) p
NegBin(r, p) th p (1 − p) Sum of r iid Geometric(p)
r probability p success r−1 p p2 1 − (1 − p)et

C OMMON C ONTINUOUS D ISTRIBUTIONS

Distribution f (x) F (x) Expected Value Variance MGF Properties

1 x−a a+b (b − a)2 ebt − eat

Uniform(a, b) Probabilities are proportional to length
b−a b−a 2 12 t(b − a)
 
2 1 (x−µ)2
− 2σ2 x−µ 2 µt+σ 2 t2 /2 µ σ2
N (µ, σ ) √ e Φ µ σ e Approximates sum of n iid rv’s w/ mean n and variance n
σ 2π σ

−λx −λx 1 1 λ
Exp(λ) λe 1−e Only memoryless continuous distribution
λ λ2 λ−t
−λx α−1
 α
λe (λx) α α λ
Gamma(α, λ) ugly Sum of α independent Exp(λ) for integer α > 0
Γ(α) λ λ2 λ−t
 Exam 1/P Formula Sheets
WWW.P ROBABILITY E XAM . COM

D ISCRETE D ISTRIBUTIONS C ONT. C ONTINUOUS D ISTRIBUTIONS C ONT. M OMENT G ENERATING F UNCTIONS

Bernoulli Uniform Definition
A Bernoulli r.v. is also an indicator of the occurrence of A ⊆ S Us ∼ Uniform(0, 1) =⇒ U = (b − a) · Us + a ∼ Uniform(a, b) tX
MX (t) = E[e ]
d−c Properties
(c, d) ⊆ (a, b) =⇒ P (c ≤ U ≤ d) = ,
X ∼ Bernoulli(p) =⇒ Y = (a − b)X + b takes values a and b b−a
with probabilities p and 1 − p, respectively. Uniquely determines a distribution
U | U ∈ (c, d) ∼ Uniform(c, d)
n (n)
E[Y ] = p · a + (1 − p) · b Normal E[X ] = MX (0)
2 Z ∼ N (0, 1) =⇒ X = σZ + µ ∼ N (µ, σ 2 ) MaX+b (t) = ebt MX (at)
Var(Y ) = (a − b) · p · (1 − p)
Φ(−z) = 1 − Φ(z) X, Y independent =⇒ MX+Y (t) = MX (t) · MY (t)
Binomial
X∼ 2
N (µX , σX ), Y ∼ 2
N (µY , σY ) independent, then Variance Shortcut (Cumulant)
If X ∼ Binomial(n, p), Y ∼ Binomial(m, p) then
d2
2

X + Y ∼ N (µX + µY , σX + σY2 ) 
n − X ∼ Binomial(n, 1 − p) Var(X) = 2 ln (M (t))
dt t=0
Central Limit Theorem
X, Y independent =⇒ X + Y ∼ Binomial(n + m, p)
X1 , . . . , Xn iid each with mean µ and variance σ 2 , then
Z | X ∼ Binomial(X, q) =⇒ Z ∼ Binomial(n, pq) 2 P ROBABILITY G ENERATING F UNCTIONS
X1 + · · · + Xn ∼
˙ N (nµ, nσ )
Poisson Defined for nonnegative integer-valued random variables
Approximating consecutive integer-valued X with CLT: Definition
If X ∼ Poisson(λ), Y ∼ Poisson(κ) then
1 1
x+ 2 −µ x− 2 −µ
   
X
λ P (X ≤ x) ≈ Φ , P (X < x) ≈ Φ GX (t) = E[t ]
P (X = x + 1) = P (X = x) · σ σ
x+1 Properties
λ ∈ N =⇒ λ, λ + 1 are modes Exponential
Uniquely determines a distribution
−λa −λb
X, Y independent =⇒ X + Y ∼ Poisson(λ + κ) X ∼ Exp(λ) =⇒ P (a ≤ X ≤ b) = e −e (k)
P (X = k) = GX (0)/k!
Z | X ∼ Binomial(X, p) =⇒ Z ∼ Poisson(λ · p) The continuous analog of the Geometric (rounding & limit) b a
GaX+b (t) = t GX (t )
P (X ≥ s + t | X ≥ s) = P (X ≥ t) [memoryless]
Geometric X, Y independent =⇒ GX+Y (t) = GX (t) · GY (t)
st Time between events in Poisson process with rate λ
If X ∼ Geometric(p) then X + 1 counts trials until 1 success Moments
1 1−p Xi ∼ Exp(λi ) indep. =⇒ min{X1 , . . . , Xn } ∼ Exp(λ1 + · · · + λn )  
E[X + 1] = Var(X + 1) = (n) X!
p p2 GX (1) =E = E[X(X − 1) . . . (X − n + 1)]
Gamma (X − n)!
P (X ≥ n + m | X ≥ m) = P (X ≥ n) [memoryless] 0
With integer α, X ∼ Gamma(α, λ) is E[X] = GX (1)
a sum of α independent Exp(λ) Var(X) = G00X (1) − (G0X (1))2 + G0X (1)
th
the time until α event in a Poisson process with rate λ

The continuous analog of the Negative Binomial

 Exam 1/P Formula Sheets
WWW.P ROBABILITY E XAM . COM

J OINT D ISTRIBUTIONS J OINT D ISTRIBUTIONS C ONT.

Cumulative Distribution Function (CDF) Covariance
F (x, y) = P (X ≤ x, Y ≤ y) Definition: Cov(X, Y ) = E[(X − µX ) · (Y − µY )] = E[XY ] − E[X] · E[Y ]
Probability Mass Function (PMF) Probability Density Function (PDF)
Cov(X, X) = Var(X) Cov(X, Y ) = Cov(Y, X) Cov(X + c, Y ) = Cov(X, Y )
∂2
p(x, y) = P (X = x, Y = y) f (x, y) = F (x, y) Cov(X, c) = 0 Cov(cX, Y ) = c · Cov(X, Y ) Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z)
∂x∂y
2 2
Var(aX + bY ) = a · Var(X) + b · Var(Y ) + 2ab · Cov(X, Y )
Marginal PMF Marginal PDF
Z ∞ Coefficient of Correlation
X
pX (x) = p(x, y) fX (x) = f (x, y) dy
 
X − µX Y − µY Cov(X, Y )
y −∞ ρX,Y = Cov , = −1 ≤ ρX,Y ≤ 1
σX σY σX · σY
Conditional PMF Conditional PDF Consequences of Independence
p(x, y) f (x, y)
pX|Y (x | Y = y) = fX|Y (x | Y = y) = E[g(X) · h(Y )] = E[g(X)] · E[h(Y )] Cov(X, Y ) = 0
pY (y) fY (y)
MX,Y (s, t) = MX (s) · MY (t) ρX,Y = 0
Independence Criteria (X and Y are independent if any hold)
Bivariate Continuous Uniform
F (x, y) = FX (x) · FY (y) − ∞ < x, y < ∞ F (x, y) = FX (x) · FY (y) − ∞ < x, y < ∞
1
p(x, y) = pX (x) · pY (y) − ∞ < x, y < ∞ f (x, y) = fX (x) · fY (y) − ∞ < x, y < ∞ f (x, y) = Probabilities are proportional to areas
Area of support
pX|Y (x | Y = y) = pX (x) where pY (y) > 0 fX|Y (x | Y = y) = fX (x) where fY (y) > 0
Multinomial
pY |X (y | X = x) = pY (y) where pX (x) > 0 fY |X (y | X = x) = fY (y) where fX (x) > 0
p(x, y) = g(x) · h(y) for any g, h ≥ 0 f (x, y) = g(x) · h(y) for any g, h ≥ 0 (X1 , . . . , Xk ) ∼ Multinomial(n, p1 , . . . , pk ) if n indep. trials performed, each with k possible outcomes
(with respective probabilities p1 , . . . , pk ) and Xi is the number of trials resulting in outcome i.
2D LOTUS
∞ ∞ n! n1 n2 nk
P (X1 = n1 , . . . , Xk = nk ) = · p1 p2 · · · pk
XX Z Z
E[h(X, Y )] = h(x, y) · p(x, y) E[h(X, Y )] = h(x, y) · f (x, y) dx dy n1 !n2 ! . . . nk !
x y −∞ −∞
Xi ∼ Binomial(n, pi ) i 6= j =⇒ Cov(Xi , Xj ) = −npi pj
Conditional Expectation Bivariate Normal
Z ∞ 2 2
(X, Y ) ∼ BivNormal(µX , µY , σX , σY , ρX,Y ) if aX + bY is normal for all a, b ∈ R.
X
E[X | Y = y] = x · pX|Y (x | Y = y) E[X | Y = y] = x · fX|Y (x | Y = y) dx
x −∞ σY 2 2
Y | X = x ∼ N (µY + ρ (x − µX ), σY (1 − ρ ))
σX
Law of Total Expectation/Variance

E[X] = E[E[X | Y ]] Var(X) = E[Var(X | Y )] + Var(E[X | Y ])

Joint Moment Generating Function
n+m

sX+tY n m ∂ 
MX,Y (s, t) = E[e ] E[X Y ] = n m MX,Y (s, t) MX (s) = MX,Y (s, 0)
∂s ∂t s=t=0
 Exam 1/P Formula Sheets
WWW.P ROBABILITY E XAM . COM

T RANSFORMATIONS T RANSFORMATIONS C ONT. R ISK AND I NSURANCE

Transformation of Discrete X Convolution Theorem (X and Y independent) Ordinary Deductible d
X Z ∞ (
P (g(X) = y) = P (X = x) X
0, X≤d
pX+Y (t) = pX (x)·pY (t−x) fX+Y (t) = fX (x)·fY (t−x) dx Payment: (X − d)+ =
x|g(x)=y
x −∞ X − d, X>d
Transformation of Continuous X Z ∞ Z ∞
Order Statistics Mean: (x − d) · fX (x) dx = SX (x) dx
Find sets Ay so g(X) ≤ y ⇐⇒ X ∈ Ay th d d
X(i) is the i smallest of iid continuous X1 , . . . , Xn w/ dist. F, f  
Compute FY (y) = P (g(X) ≤ y) = P (X ∈ Ay ) d
n
X n   Payment w/ inflation r: (1 + r) · X −
fY (y) = FY0 (y) FX(j) (x) = k
F (x) (1 − F (x)) n−k 1+r +
k
k=j
Strictly Monotone Transformation of Continuous X   Policy Limit u
n−1
fX(j) (x) = n f (x)F (x)j−1 (1 − F (x))n−j
(
fY (y) = fX (g −1 (y)) · |(g −1 )0 (y)| j−1 X, X≤u
Payment: X ∧ u =
u, X>u
1-1 Transformation of Continuous X, Y
n n
( ( FX(1) (x) = 1 − (1 − F (x)) FX(n) (x) = F (x) Z u Z u
g1 (x, y) = u x = h1 (u, v) Mean: x · fX (x) dx + u · SX (u) = SX (x) dx
If has unique solution and Mixture Distribution 0 0
g2 (x, y) = v y = h2 (u, v)  
X has PMF/PDF f (x) = α1 · f1 (x) + · · · + αn · fn (x) u
 ∂h
1 ∂ h1 
 Payment w/ inflation r: (1 + r) · X ∧

 ∂u ∂v  ∂ h1 ∂ h2 ∂ h1 ∂ h2 1+r
J= = − 6= 0 (αi positive and sum to 1, fi are valid PMF/PDFs)
∂u ∂v ∂v ∂u

 ∂ h2 ∂ h2 
∂u ∂v Ordinary Deductible d and Policy Limit u Simultaneously
Then the joint density of U = g1 (X, Y ) and V = g2 (X, Y ) is FX (x) = α1 · F1 (x) + · · · + αn · Fn (x)

0,
 X≤d
g(u, v) = f (h1 (u, v), h2 (u, v)) · |J| SX (x) = α1 · S1 (x) + · · · + αn · Sn (x) Payment: Xdu = X − d, d < X ≤ u + d

k k k u, X >u+d

E[X ] = α1 · E[X1 ] + · · · + αn · E[Xn ]
MX (t) = α1 · MX1 (t) + · · · + αn · MXn (t)
Z u+d Z u+d
Mean: (x − d) · fX (x) dx + u · SX (u + d) = SX (x) dx
d d
u
1+r
Payment w/ inflation r: (1 + r) · X d
1+r

Loss Given Positive Loss

If XC = X | X > 0 and α = P (X > 0), then
2 2
E[X] = α · E[XC ] E[X ] = α · E[XC ]
2
Var(X) = α · Var(XC ) + α · (1 − α) · E[XC ]
 Mark’s Formula Sheet for Exam P
Discrete distributions

• Uniform, U (m)
1
– PMF: f (x) = m, for x = 1, 2, . . . , m
m+1 m2 − 1
– µ= and σ 2 =
2 12
• Hypergeometric
N1 N2



x n−x
– PMF: f (x) = N


n
– x is the number of items from the sample of n items that are from group/type 1.
N1 N1 N2 N − n
– µ = n( ) and σ 2 = n( )( )( )
N N N N −1
• Binomial, b(n, p)
 
n x
– PMF: f (x) = p (1 − p)n−x , for x = 0, 1, . . . , n
x
– x is the number of successes in n trials.
– µ = np and σ 2 = np(1 − p) = npq
– MGF: M (t) = [(1 − p) + pet ]n = (q + pet )n

• Negative Binomial, nb(r, p)

 
x−1 r
– PMF: f (x) = p (1 − p)x−r , for x = r, r + 1, r + 2, . . .
r−1
– x is the number of trials necessary to see r successes.
1 r r(1 − p) rq
– µ = r( ) = and σ 2 = 2
= 2
p p p p
t r
r
pet

(pe )
– MGF: M (t) = =
[1 − (1 − p)et ]r 1 − qet
• Geometric, geo(p)

– PMF: f (x) = (1 − p)x−1 p, for x = 1, 2, . . .

– x is the number of trials necessary to see 1 success.
– CDF: P (X ≤ k) = 1 − (1 − p)k = 1 − q k and P (X > k) = (1 − p)k = q k
1 1−p q
– µ = and σ 2 = 2
= 2
p p p
pe t pet
– MGF: M (t) = =
1 − (1 − p)et 1 − qet
– Distribution is said to be “memoryless”, because P (X > k + j|X > k) = P (X > j).
 • Poisson
λx e−λ
– PMF: f (x) = , for x = 0, 1, 2, . . .
x!
– x is the number of changes in a unit of time or length.
– λ is the average number of changes in a unit of time or length in a Poisson process.
λ2 λx
– CDF: P (X ≤ x) = e−λ (1 + λ + 2! + ··· + x! )
– µ = σ2 = λ
t −1)
– MGF: M (t) = eλ(e

Continuous Distributions
• Uniform, U (a, b)
1
– PDF: f (x) = , for a ≤ x ≤ b
b−a
x−a
– CDF: P (X ≤ x) = , for a ≤ x ≤ b
b−a
a+b (b − a)2
– µ= and σ 2 =
2 12
etb − eta
– MGF: M (t) = , for t 6= 0, and M (0) = 1
t(b − a)
• Exponential
1
– PDF: f (x) = e−x/θ , for x ≥ 0
θ
– x is the waiting time we are experiencing to see one change occur.
– θ is the average waiting time between changes in a Poisson process. (Sometimes called the
“hazard rate”.)
– CDF: P (X ≤ x) = 1 − e−x/θ , for x ≥ 0.
– µ = θ and σ 2 = θ2
1
– MGF: M (t) =
1 − θt
– Distribution is said to be “memoryless”, because P (X ≥ x1 + x2 |X ≥ x1 ) = P (X ≥ x2 ).

• Gamma
1 1
– PDF: f (x) = α
xα−1 e−x/θ = xα−1 e−x/θ , for x ≥ 0
Γ(α)θ (α − 1)!θα
– x is the waiting time we are experiencing to see α changes.
– θ is the average waiting time between changes in a Poisson process and α is the number of
changes that we are waiting to see.
– µ = αθ and σ 2 = αθ2
1
– MGF: M (t) =
(1 − θt)α
• Chi-square (Gamma with θ = 2 and α = 2r )
1
– PDF: f (x) = xr/2−1 e−x/2 , for x ≥ 0
Γ(r/2)2r/2
– µ = r and σ 2 = 2r
1
– MGF: M (t) =
(1 − 2t)r/2
 • Normal, N (µ, σ 2 )
1 2 2
– PDF: f (x) = √ e−(x−µ) /2σ
σ 2π
2 t2 /2
– MGF: M (t) = eµt+σ

Integration formulas
Z
1 1 1
• p(x)eax dx = p(x)eax − 2 p0 (x)eax + 3 p00 (x)eax − . . .
a a a
Z ∞  
1 −x/θ
• x e dx = (a + θ)e−a/θ
a θ
Z ∞  
2 1 −x/θ
• x e dx = ((a + θ)2 + θ2 )e−a/θ
a θ
Other Useful Facts

• σ 2 = E[(X − µ)2 ] = E[X 2 ] − µ2 = M 00 (0) − M 0 (0)2

• Cov(X, Y ) = E[(X − µx )(Y − µy )] = E[XY ] − µx µy

• Cov(X, Y ) = σxy = ρσx σy

and
σxy
ρ=
σx σy
σy
• Least squares regression line: y = µy + ρ (x − µx )
σx
• When variables X1 , X2 , . . . , Xn are not pairwise independent, then
Xn Xn X
Var( Xi ) = σi2 + 2 σij
i=1 i=1 i<j
and
Xn n
X X
Var( ai Xi ) = a2i σi2 + 2 ai aj σij
i=1 i=1 i<j
where σij is the covariance of Xi and Xj .

• When X depends upon Y , E[X] = E[E[X|Y ]].

• When X depends upon Y , Var(X) = E[Var(X|Y )] + Var(E[X|Y ]). (Called the “Total Variance” of
X.)

• Chebyshev’s Inequality: For a random variable X having any distribution with finite mean µ and
variance σ 2 , P (|X − µ| ≥ kσ) ≤ k12 .

• For the variables X and Y having the joint PMF/PDF f (x, y), the moment generating function for
this distribution is
XX
M (t1 , t2 ) = E[et1 X+t2 Y ] = E[et1 X et2 Y ] = et1 x et2 y f (x, y)
x y

– µx = Mt1 (0, 0) and µy = Mt2 (0, 0) (These are the first partial derivatives.)
– E[X 2 ] = Mt1 t1 (0, 0) and E[Y 2 ] = Mt2 t2 (0, 0) (These are the “pure” second partial derivatives.)
– E[XY ] = Mt1 t2 (0, 0) = Mt2 t1 (0, 0) (These are the “mixed” second partial derivatives.)

• Central Limit Theorem: As the sample size n grows,

 n
X
– the distribution of Xi becomes approximately normal with mean nµ and variance nσ 2
i=1
n
1X σ2
– the distribution of X̄ = Xi becomes approximately normal with mean µ and variance .
n n
i=1

• If X and Y are joint distributed with PMF f (x, y), then

X
– the marginal distribution of X is given by fx (x) = f (x, y)
y
X
– the marginal distribution of Y is given by fy (y) = f (x, y)
x
f (x, y0 )
– f (x|y = y0 ) = .
fy (y0 )
P
X xf (x, y0 )
– E[X|Y = y0 ] = xf (x|y = y0 ) = x
x
fy (y0 )
 Exam Cheat Sheet
Binomial Coefficient: For r ≤ n,  
n n!
=
m m!(n − m)!
represents the number of possible combinations of n objects taken m at a time.
Multinomial Coefficient: For n1 + n2 + · · · + nr = n,
 
n n!
=
n1 , n2 , ..., nr n1 !n2 ! · · · nr !
represents the number of possible ways n objects can be divided into r groups with
n1 , n2 , ..., nr objects in each, respectively.
Axioms for Probability Measures: The following 3 axioms define a probability measure:

1. For all events E, 0 ≤ P (E) ≤ 1.

2. If S is the sample space, P (S) = 1.
P∞
i=1 Ei ) =
3. For any sequence of mutually exclusive events E1 , E2 , ..., En , P (∪∞ i=1 P (Ei ).

Useful Formulas for Probability Measures:

P (E c ) = 1 − P (E)
P (E) ≤ P (F ) if E ⊆ F
P (E) = P (EF ) if E ⊆ F
P (E ∪ F ) = P (E) + P (F ) − P (EF )
P (E) = P (EF ) + P (EF c )
P (E ∪ F ∪ G) = P (E) + P (F ) + P (G) − P (EF ) − P (EG) − P (F G) + P (EF G)

Conditional Probability: The probability of event E, given that we observe F , is:

P (EF )
P (E | F ) =
P (F )
Also, P (E | F ) defines a new probability measure with F being the restriced sample space.
Independence: Two items are independent if P (EF ) = P (E)P (F ). If P (E | F ) = P (E), E and
F are independent; when P (F ) 6= 0, the two definitions are exactly equivalent.
Useful Formulas for Contitional Probability:
P (EF ) = P (E | F )P (F ) = P (F | E)P (E)
P (E) = P (E | F )P (F ) + P (E | F c )P (F c )
P (E1 E2 · · · En ) = P (E1 )P (E2 | E1 )P (E3 | E2 E1 ) · · · P (En | En−1 En−2 · · · E1 )
= P (En )P (En−1 | En )P (En−2 | En−1 En ) · · · P (E1 | E2 E3 · · · En )
P (F | E)P (E)
P (E | F ) =
P (F )
Expectation – Discrete: Suppose p(x) = P(X = x) is the probability mass function of a R.V.
X. Then
X
EX = xp(x)
Xx
E [g(X)] = g(x)p(x)
x
Var [X] = E (X − µ)2 = E X 2 − ( E X)2 , where µ = E X
  

Discrete Distributions:

If X ∼ Bernoulli(x; p), where p is the probability of success, then


 p x=1
P(X = x) = 1−p x=0 , E X = p, Var [X] = p(1 − p)
0 otherwise


If X ∼ Binomial (x; n, p), where p is the prob. of success and n is the number of tries, then

n
px (1 − p)n−x x ∈ {0, 1, ..., n}


P(X = x) = x , E X = np, Var [X] = np(1 − p)
0 otherwise

If X ∼ Geometric(x; p), then

(1 − p)x−1 p x ∈ {0, 1, ...}


1−p
P(X = x) = , E X = 1/p Var [X] =
0 otherwise p2

If X ∼ NegativeBinomial (x; r, p), where p is the prob. of success, and r is the number of
successes required,

x−1
pr (1 − p)x−r x ∈ {r, r + 1, ...}


r−1
r
P(X = x) = EX =
0 otherwise p

If X ∼ Poisson(x; λ), where λ is the rate parameter, then

e−λ λx

x ∈ {0, 1, 2, ...}
P(X = x) = x! EX = λ Var [X] = λ
0 otherwise

If X ∼ HyperGeometric(x; n, N, m), then X represents the number of red balls in a sample of n

balls drawn from an urn with N balls, m of which are red. Then
( h
i h i
m N −m N −1



x n−x n x ∈ {0, 1, ..., n} nm
P(X = x) = EX =
0 otherwise N

2
Expectation: Continuous case. Suppose fX (x) is the probability density function of X. Then
ˆ ∞
EX = xfX (x)dx
ˆ−∞
∞
E [g(X)] = g(x)fX (x)dx
−∞
Var [X] = E (X − µ)2 = E X 2 − ( E X)2 , where µ = E X
   

Continuous distributions:

If X ∼ Uniform(x; a, b), with a < b, then

1 (b − a)2

b−a x ∈ [a, b] b+a
fX (x) = EX = Var [X] =
0 otherwise 2 12

If X ∼ Exponential (x; λ), with λ being the rate, then


λe−λx x ≥ 0 1 1
fX (x) = EX = Var [X] =
0 otherwise λ λ2

If X ∼ N x; µ, σ 2 , where µ is the expectation and σ 2 is the variance, then

1 2 2
fX (x) = √ e−(x−µ) /2σ EX = µ Var [X] = σ 2
2π σ

Standard Normal CDF: If Z ∼ N (z; 0, 1), then define Φ(z) = P (Z ≤ z).

Poisson Approximation Theorem: Let X ∼ Binomial (x; n, p). For np small relative to n,
P(X = x) ≃ P(Y = x), where Y ∼ Poisson(y; λ = np).

DeMoivre-Laplace Approximation Theorem: Let X ∼ Binomial (x; n, p). For np(1 − p) suffi-
ciently large, !
X − np
P p ≤ z ≃ P(Z ≤ z) = Φ(z)
np(1 − p)
where Z ∼ N (z; 0, 1).