P-exam

Thingy

P(A n B) = *using (#|#)*

P(A n B) = P(A|B)*P(B) = P(B|A)*P(A)

P(A n B) = *using (# n #)*

P(A n B) = P(A) - P(A n B’) = P(B) - P(B n A’)

P(A) = *using (#|#)*

P(A) = P(A|B)*P(B) + P(A|B’)*P(B’)

P(A|B) = *using compliments*

P(A|B) = 1-P(A’|B)

P(A|B) = *Bayes Formula*

P(A|B) = P(B|A)*P(A) / P(B|A)*P(A)+P(B|A’)*P(A’)

(n k) *choose formula*

(n k) = n! / k!(n-k)!

ln(e^x) =

ln(e^x) = x

E(X) =

E(X) = SUM(x_i * p_i)

E(X) = ∫_−∞^∞​x*f(x) dx

E(X²) =

E(X²) = Var(X) + E(X)²

E(X²) = SUM(x_i² * p_i)

E(aX + bX²) =

E(aX + bX²) = a*E(X) + b*E(X²)

Var(X) =

Var(X) = E(X²) - E(X)²

Var(Y|X) =

Var(Y|X) = E(Y²|X) - E(Y|X)²

Given P(A|B) = k*P(A|B’), what is P(B|A)?

k*P(B) / (k-1)*P(B)-1

Var(X+Y) =

Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)

Cov(X,Y) =

Cov(X,Y) = E(XY) - E(X)*E(Y)

p *roe*

p = Cov(X,Y) / sqrt( Var(X)*Var(Y) )

Binomial Dist - pdf

(n k) * p^k *(1-p)^n-k

Binomial - E(X)

np

Binomial - Var(X)

np(1-p)

Poisson - pdf

e^-λλ^k / k!

Poisson - E(X) & Var(X)

λ

Negative Binomial - definitions “r & k & p”

r-th success ends the trial

k - # of failures “P(X=k)”

p - probability of success

Negative Binomial - pdf

(k+r-1 k) * p^r * (1-p)^k

Negative Binomial - E(X) *failures

r(1-p) / p

Negative Binomial - Var(X) *failures

r(1-p) / p²

Exponential - cdf

P(T < t) = 1-e^-(1/λ)t

Exponential - λ

1/λ = “once every λ ___(ex. years)” or “λ times every ___(ex. years)”
1/λ = mean

Exponential - E(X)

1/λ

Exponential - Var(X)

1/λ²

Beta - E(X)

α/(α+β)

Beta - Var(X)

αβ / (α+β)²(α+β+1)

Loss-RV (“d”: deductible, “c”: cap)

0 , x<d

x-d , d < x < c+d

c , x > c + d

Uniform - E(X)

a+b / 2

Uniform Var(X) *continuous

(b-a)² / 12

A*X ~ N(#, #)

A*X ~ N(A*μ, A²*σ²)

def of “e”

Σ1/n!

“Name idk what it is called lol” E[X ^ d]

θ(1-e^-d/θ)

“Name idk what it is called lol” E[X ^ u]

θ(1-e^-u/θ) =(if ‘u’/’cap’ is inf.)= θ

Expected cost per Loss

1-α(E[X ^ u] - E[X ^ d])

Uniform Var(X) *discrete

n²-1 / 12

or

(b-a+1)² -1 / 12

E(X|Y)

Σ{X_i*P(X_i|Y)}/P(Y)

Exponential - pdf

λe^−λx

*x* of group *K*, with Population *N*, and *n* picks.

(K : x)(N-K : n-x) / (N : n)

Negative Binomial - E(X) *total

r/p

Negative Binomial - Var(X) *total

r(1-p)/p²

Var(a*X - b*Y)

a²Var(X) + b²Var(Y) - a*b*Cov(X,Y)

Darth Vader Rule - E[X]

E[X] =∫₀^m(1-F(X))dx

Var(X₁ + X₂ + … + X_n) = *X’s are IID*

n*Var(X)

Expected value of a deductible claim

(Y= X-a, X>a, u→cap, a→deductible)
E[Y] = ∫_a^u (x-a)_*f(x) dx + (u-d)_*P(X>a)
or
(if… f(x)=e^x/θ)
E[Y] = θ(e^-a/θ - e^-u/θ)