SOA Exam P Questions with correct Answers

SOA Exam P Questions with correct
Answers

P(A) - Correct Answers -i) 1-P(A')
ii) (A∩B)+(A∩B')
iii) (A∪B)-(A'∩B)

Gamma distribution
E(X) - Correct Answers -α/β

Gamma distribution
Var(X) - Correct Answers -α/β²

P(a<X≤b) - Correct Answers -F(b)-F(a)

Probability mass - Correct Answers -probability of a discrete r.v. at a point

E(X+Y) - Correct Answers -E(X)+E(Y)

Conditional probabilities
P(A) - Correct Answers -P(A|B)P(B)+P(A|B')P(B')

Exponential distribution
Moment Generating Function - Correct Answers -λ/(λ-t)

Taylor Series expansion of e^x
is what summation - Correct Answers -(0,∞) ∑ (x^k)/k!

Find P(X=Y) when X and Y are different distributions - Correct Answers -
P(X=Y)=P(X=k)*P(Y=k)
= (0 to ∞)∑ P(X=k)*P(Y=k)
use Taylor series to simplify

Given distribution of X and Y=u(X) then
fy(y)= - Correct Answers -fy(y)= fx(v(y)) *|v'(y)|
where X=v(y)

Using rho
Cov(X,Y)= - Correct Answers -(rho)σxσy

For integration by parts that involve e^-x: - Correct Answers -dv=e^-x

,Exponential distribution
P(a<x≤b)= - Correct Answers -e^(-λa)-e^(-λb)

How many cards are in a deck?
How many ranks?
How many possible pairs of a rank? - Correct Answers -52 cards
13 ranks
(4 chose 2) = 6 possible pairs

Geometric distribution random variable X: - Correct Answers -X: the number of failures
until the first success

M(t₁,0) = - Correct Answers -Mx(t₁)

Given discrete λ,
P(X=n+1) = - Correct Answers -P(X=n) * (λ/(n+1))

Order Statistic - Correct Answers -ordering a random sample of X's using another r.v. Y

Continuous uniform
probability density function - Correct Answers -1/(b-a)

Continuous uniform
cumulative distribution function - Correct Answers -0 X<a
(x-a)/(b-a)
1 X>b

Var( aX+bY+c) - Correct Answers -a²var(x)+
b²var(Y)+2abCov(X,Y)

Geometric distribution
probability function - Correct Answers -pq^x

Binomial distribution
moment generating function - Correct Answers -(pe^t+q)ⁿ

General increasing geo series - Correct Answers -1+2r+3r²+4r³+....
= 1/(1-r)²

Geometric distribution
E(X)= - Correct Answers -q/p

Geometric distribution
Var(X)= - Correct Answers -q/p²

, Geometric progression
sum of the first n terms - Correct Answers -a+ar+ar²+ar³+ar^(n-1)
= a((1-rⁿ)/(1-r))

Geometric progression
infinite sum - Correct Answers -a/(1-r)

When is exponential distribution used? - Correct Answers -used to model the time until
an event occurs

If Y=aX+b then
My(t)= - Correct Answers -e^(bt)*Mx(at)

Given geometric distribution
P(X=n+k|X=≥n)= - Correct Answers -P(X=k)

Mixture of distributions of X₁,X₂
probability density function f(x)= - Correct Answers -f(x)= af₁(x)+(1-a)f₂(x)

Proportional insurance - Correct Answers -When the insurer pays αX of the loss where
0<α<1

Joint uniform distribution over area R
probability density function - Correct Answers -f(x,y)=1/(area of R)

Joint uniform distribution over area R
P(A)= - Correct Answers -P(A)=
(area of A)/(area of R)

The number of ways to order n objects where
n₁ are type 1
n₂ are type 2... - Correct Answers -n!
n₁!n₂!...!nt

Cov(X,Y) - Correct Answers -E(XY)-E(X)E(Y)

Cov(X,X) - Correct Answers -Var(X)

f(x|Y=y) - Correct Answers -f(x,y)/(fy(y))

P(max(X₁,X₂,X₃)≤y) - Correct Answers -P(X₁≤y)∩P(X₂≤y)∩
P(X₃≤y)

P(min(X₁,X₂,X₃)>y) - Correct Answers -P(X₁>y)∩P(X₂>y)∩
P(X₃>y)