SOA (Society of Actuaries) Exams Actuarial exams (ASA, FSA levels) SOA Exam P 4 SETS COMBINED SOA Exam P/ SOA Final Exam/ SOA EXAM STAM Detailed Solutions 100% WITH COMPLETE SOLUTION NEW!!

SOA (Society of Actuaries) Exams

Actuarial exams (ASA, FSA levels)
SOA Exam P 4 SETS COMBINED SOA Exam P/ SOA Final
Exam/ SOA EXAM STAM Detailed Solu&ons 100% WITH
COMPLETE SOLUTION NEW!!

Bernoulli Trial - An experiment in which there are exactly two possible outcomes



Binomial distribu on - A random variable X represents the number of successes observed from
the n Bernoulli trials



Binomial Parameters - n = number of trials

p = probability of success

q = 1-p



Binomial probability func on (f(x)) -



E[X] binomial func on - E[x] = np



var(x) binomial func on - var(x) = npq



MGF binomial func on -



Addi ve property of binomial func on - Sum of independent binomially distributed variables
each with probability p, has parameters of p and the sum of all n

, Nega ve binomial distribu on - X is the number of failures before r successes in a series of
independent Bernoulli trials



Nega ve binomial parameters - r = desired number of successes

x = number of failures before r successes

p = probability of success

q = 1-p, probability of failure



Nega ve binomial probability density func on f(x) - f(x) = Pr(X=x) = (r + x - 1)!/x!(r-1)! * p^r *
q^x



Expected value of nega ve binomial distribu on E[X] - E[X] = rq/p



Variance of nega ve binomial distribu on var(x) - var(x) = rq/p^2



Moment genera ng func on nega ve binomial distribu on - Mx(t) = ((1 - qe^t)/p)^-r



Addi ve property of nega ve binomial distribu on - If Xi follows a nega ve binomial
distribu on with parameters ri and p, and they are independent, then the sum of them follows
a nega ve binomial distribu on with parameters as the sum of the ri and p.



Geometric distribu on - The number of failures observed from the series of Bernoulli trials un l
the first success occurs



Parameters of geometric distribu on - p = probability of success

q = 1-p

,x = number of trials before first success



Probability density func on of geometric distribu on - f(x) = Pr(X=x) = q^x * p



Probability mass func on of geometric distribu on - F(x) = Pr(X <= x) = 1 - q^x+1



Expected value of geometric distribu on - E[x] = q/p



Variance of geometric distribu on - var(x) = q/p^2



MGF of geometric distribu on - Mx(t) = ((1 - qe^t)/p)^-1



Addi ve property of geometric distribu on - A sum of n independent geometric distribu ons
with parameter p follows a nega ve binomial distribu on with parameters r = n and p.



Hypergeometric distribu on defini on - X is the number of Type 1 objects in a sample of n
objects randomly selected without replacement from a popula on of m objects, of which m1
are type 1 and m2 are type 2 (where m = m1 + m2.



Hypergeometric probability distribu on func on - f(x) = Pr(X=x) = C(m1,x) * C(m2,n-x) = C(m,n)



Expected value of hypergeometric func on - E[x] = nm1/m



Variance of hypergeometric func on - var(x) = n(m1/m)(m2/m)((m-n)/(m-1))



Defini on of poisson distribu on - X is the number of occurrences of some "rare" event in a
unit me period where λ is the rate of occurrences per unit me period

, Probability distribu on func on - f(x) = Pr(X=x) = e^-λ*λ^x/x!



Expected value of a poisson distribu on - λ



Variance of poisson distribu on - λ



MGF of poisson distribu on - Mx(t) = e^(λ((e^t)-1))



Addi ve property of poisson distribu on - Sum of poisson distribu ons has parameter of the
sum of all the λ



Uniform distribu on - Constant probability density func on on the interval [a,b]



PDF of uniform distribu on - f(x) = 1/(b-a)



CDF of uniform distribu on - F(x) = (x-a)/(b-a)



Expected value of uniform distribu on - E(x) = (b+a)/2



Variance of uniform distribu on - var(x) = ((b-a)^2)/12



median of uniform distribu on - m = (b+a)/2



MGF of uniform distribu on - Mx(t) = (e^bt - e^at)/t(b-a)