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Actuary Exam P formula Sheet, Cheat Sheet of Mathematics

Formula Sheet With Discrete and Continous Distributions,Integration Formulas and Other Useful Facts.

Typology: Cheat Sheet

2021/2022

Uploaded on 02/07/2022

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Mark’s Formula Sheet for Exam P
Discrete distributions
Uniform, U(m)
PMF: f(x) = 1
m, for x= 1,2, . . . , m
µ=m+ 1
2and σ2=m21
12
Hypergeometric
PMF: f(x) = N1
xN2
nx
N
n
xis the number of items from the sample of nitems that are from group/type 1.
µ=n(N1
N) and σ2=n(N1
N)(N2
N)(Nn
N1)
Binomial, b(n, p)
PMF: f(x) = n
xpx(1 p)nx, for x= 0,1, . . . , n
xis the number of successes in ntrials.
µ=np and σ2=np(1 p) = npq
MGF: M(t) = [(1 p) + pet]n= (q+pet)n
Negative Binomial, nb(r, p)
PMF: f(x) = x1
r1pr(1 p)xr, for x=r, r + 1, r + 2, . . .
xis the number of trials necessary to see rsuccesses.
µ=r(1
p) = r
pand σ2=r(1 p)
p2=rq
p2
MGF: M(t) = (pet)r
[1 (1 p)et]r=pet
1qetr
Geometric, geo(p)
PMF: f(x) = (1 p)x1p, for x= 1,2, . . .
xis the number of trials necessary to see 1 success.
CDF: P(Xk)=1(1 p)k= 1 qkand P(X > k) = (1 p)k=qk
µ=1
pand σ2=1p
p2=q
p2
MGF: M(t) = pet
1(1 p)et=pet
1qet
Distribution is said to be “memoryless”, because P(X > k +j|X > k) = P(X > j).
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Mark’s Formula Sheet for Exam P

Discrete distributions

  • Uniform, U (m)
    • PMF: f (x) = (^) m^1 , for x = 1, 2 ,... , m
    • μ = m^ + 1 2

and σ^2 = m

  • Hypergeometric
    • PMF: f (x) =

(N 1

x

)( N 2

n−x

(N

n

  • x is the number of items from the sample of n items that are from group/type 1.
  • μ = n( N N^1 ) and σ^2 = n( N N^1 )( N N^2 )( NN^ −−^ n 1 )
  • Binomial, b(n, p)
  • PMF: f (x) =

n x

px(1 − p)n−x, for x = 0, 1 ,... , n

  • 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)
  • PMF: f (x) =

x − 1 r − 1

pr(1 − p)x−r, for x = r, r + 1, r + 2,...

  • x is the number of trials necessary to see r successes.
  • μ = r(

p ) =^

r p and^ σ

(^2) = r(1^ −^ p) p^2 =^

rq p^2

  • MGF: M (t) =

(pet)r [1 − (1 − p)et]r^ =

pet 1 − qet

)r

  • 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 − qk^ and P (X > k) = (1 − p)k^ = qk
    • μ =

p and^ σ

(^2) =^1 −^ p p^2 =^

q p^2

  • MGF: M (t) =

pet 1 − (1 − p)et^ =^

pet 1 − qet

  • Distribution is said to be “memoryless”, because P (X > k + j|X > k) = P (X > j).
  • Poisson
    • PMF: f (x) = λ

xe−λ x! , for^ x^ = 0,^1 ,^2 ,...

  • 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.
  • CDF: P (X ≤ x) = e−λ(1 + λ + λ 2!^2 + · · · + λ xx! )
  • μ = σ^2 = λ
  • MGF: M (t) = eλ(et−1)

Continuous Distributions

  • Uniform, U (a, b)
    • PDF: f (x) = 1 b − a

, for a ≤ x ≤ b

  • CDF: P (X ≤ x) = xb −− aa , for a ≤ x ≤ b
  • μ = a^ +^ b 2

and σ^2 = (b^ −^ a)

2 12

  • MGF: M (t) = e

tb (^) − eta t(b − a)

, for t 6 = 0, and M (0) = 1

  • Exponential
    • PDF: f (x) =^1 θ

e−x/θ, for x ≥ 0

  • 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 = θ^2
  • MGF: M (t) = 1 1 − θt
  • Distribution is said to be “memoryless”, because P (X ≥ x 1 + x 2 |X ≥ x 1 ) = P (X ≥ x 2 ).
  • Gamma
  • PDF: f (x) =

Γ(α)θα^ x

α− (^1) e−x/θ (^) = 1 (α − 1)!θα^ x

α− (^1) e−x/θ, for x ≥ 0

  • 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 = αθ^2
  • MGF: M (t) = 1 (1 − θt)α
  • Chi-square (Gamma with θ = 2 and α = r 2 )
  • PDF: f (x) = 1 Γ(r/2)2r/^2

xr/^2 −^1 e−x/^2 , for x ≥ 0

  • μ = r and σ^2 = 2r
  • MGF: M (t) =

(1 − 2 t)r/^2

  • the distribution of

∑^ n

i=

Xi becomes approximately normal with mean nμ and variance nσ^2

  • the distribution of X¯ =^1 n

∑^ n

i=

Xi becomes approximately normal with mean μ and variance σ

2 n.

  • If X and Y are joint distributed with PMF f (x, y), then
    • the marginal distribution of X is given by fx(x) =

y

f (x, y)

  • the marginal distribution of Y is given by fy(y) =

x

f (x, y)

  • f (x|y = y 0 ) = f^ (x, y^0 ) fy(y 0 )
  • E[X|Y = y 0 ] =

x

xf (x|y = y 0 ) =

x xf^ (x, y^0 ) fy(y 0 )