Download 2001 Training Program for Pension Officers

Intensive Actuarial Training for
Bulgaria
January 2007
Lecture 0 – Review on
Probability Theory
By Michael Sze, PhD, FSA, CFA
Topics Covered
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Some definitions and properties
Moment generating functions
Some common probability distributions
Conditional probability
Properties of expectations
Some Definitions and Properties
• Cumulative distribution function F(x)
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F is non-decreasing: a < b  F(a) < F(b)
Limb F(b) = 1
Lima -  F(a) = 0
F is right continuous:bnb  LimnF(bn) = b
• E[X] =  x p(x) = , where p(x) = P(X = x)
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E[g(x)] = i xi g(xi) p(xi)
E[aX + b] = a E[X] + b
E[X2] = i xi2 p(xi)
Var(X) = E[(X - )2] = E[X2] – (E[X])2
Var(a X + b) = a2 Var (X)
Moment Generating Functions
• Definition: mgf MX(t) = E[e t x]
• Properties:
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There is a 1 – 1 correspondence between f(x) and MX(t)
X, Y independent r.v.  MX+Y(t)=MX(t).MY(t)
X1 ,…,Xn indep.  M i xi (t)= i Mxi(t)
mgf for f1 + f2 + f3 = Mx1 (t) + Mx2 (t) + M x3 (t)
M’X(0) = E[X]
M(n)X(0) = E[Xn]
Some Common Discrete
Probability Distributions
• Binomial random variable (r.v.) with
parameters (n, p)
• Poisson r.v. with parameter 
• Geometric r.v. with parameter p
• Negative binomial r.v. with parameter (r, p)
Some Common Continuous
Probability Distributions
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Uniform r.v. on (a, b)
Normal r.v. with parameter (, 2)
Exponential r.v. with parameter 
Gamma r.v. with parameters (t, ), t,  > 0
Binomial r.v. B(n, p)
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n is integer, 0  p  1
Probability of getting i heads in n trials
p(i) = nCi pi qn – i
E[X] = n p
Var(X) = n p q
MX (t) = (p et + q)n
Poisson r.v. with parameter 
•  > 0, the expected number of events
• Poisson is good approximation of binomial
for large n, small p, and not too big np
• np
• p(i) = P(X = i) = e -  x (i / i!)
• E[X] = Var(X) = 
• MX (t) = exp [  (et - 1) ]
Geometric r.v with parameter p
• 0  p  1, probability of success in one trial
• Geometric r.v. is used to study the
probability of getting the success in n trials
• p(n) = P(X = n) = qn - 1 p
• E[X] = 1/p
• Var(X) = q / p2 .
• MX (t) = p et / ( 1 - q et )
Negative Binomial r.v. with
parameter r, p
• p = probability of success in each trial
• r = number of successes wanted
• Negative binomial r.v. is used to study the
probability of getting first r successes in n trials
• p(n) = P(X = n) = n - 1Cr - 1 qn - r pr .
• E[X] = r / p
• Var(X) = r q / p2
• MX (t) = [p et / ( 1 - q et )]r
Uniform r.v. on (a, b)
• a<x<b
• f(x) =  1 / (b – a) for a < x < b
 0 otherwise
• F(c) =  (c – a) / (b – a) for a < x < b
 0 otherwise
• E[X] = (a + b) / 2
• Var(X) = (b – a)2 / 12
• MX (t) = (etb - eta) / [t (b - a)]
Normal r.v. with parameters
(, 2)
• By central limit theorem, many r.v. can be
approximated by a normal distribution
• f(x) = [1/(22)] exp [ - (x - )2 / 22]
• E[X] = 
• Var(X) = 2 .
• MX (t) = exp [ t + 2 t2 /2 ]
Exponential r.v. with parameter 
• >0
• Exponential r.v. X gives the amount of waiting
time until the next event happens
• X is memoryless: P(X>s+t|X>t) = P(X>s) for all s,
t0
• f(x) =  e - x. for x  0, 0 otherwise
• F(a) = 1 - e - a
• E[X] = 1 / 
• Var(X) = 1 / 2
• MX (t) =  / (  - t )
Gamma r.v. with parameters (s,  )
• s,  > 0
• Exponential r.v. X gives the amount of waiting
time until the next s events happen
• f(x) =  e - x (x)s – 1 / (t) for x  0, 0 otherwise
• (s) = 0 e - y ys – 1 dy
• (n) = (n – 1)! , (1) = (0) = 1
• E[X] = s / 
• Var(X) = s / 2
• MX (t) = [ / (  - t )] s
Conditional Probability
• Definition:For P(F)>0, P(E|F) = P(EF)/P(F)
• Properties:
– For A1,…,An,whereAiAj = for ij (exclusive), and
Ai = S(exhaustive), then
P(B) = i P(B|Ai ) P(Ai)
– Baye’s Theorem:
For P(B)>0, P(A|B) = [P(B|A).P(A)]/P(B)
– E[X|A] = i xi P(xi |A)
– E[X| Ai ] = i E(X|Ai) P(Ai)
Properties of Expectation
• E[X + Y] = E[X] + E[Y]
• E[i Xi ] = i E[Xi ]
• If X,Y are independent, then
E[g(X) h(Y)] = E[g(X)] E[h(Y)]
• Def.: Cov(X,Y) = E[(X-E[X])(Y-E[Y])]
• Cov(X,Y) = Cov(Y,X)
• Cov(X,X) = Var(X)
• Cov(aX,Y) = a Cov(X,Y)
Properties of Expectation(continued)
• Cov(i Xi, jYj) = i j Cov(Xi,Yj)
• Var(i Xi) = iVar(Xi) + ij Cov(Xi,Yj)
• If SN = X1+…+XN is a compound process
– Xi are mutually independent,
– Xi are independent of N, and
– Xi have the same distribution, then
E[SN] = i E[Xi]
Var(SN) = E[N] Var(X) + Var(N) (E[X])2