Risk and uncertainty - formule

Probability of Finding X in an Interval

Probability of one success or one "failure". How many failures and how many successes before 1 success.

Poisson Distribution:

• Probability function: Pr(X=k) = e^(-λ) * (λ^k) / k!
• Mean: E(X) = λ
• Variance: V(X) = λ

Binomial Random Variable:

• Probability function: Pr(X=k) = (n choose k) * p^k * (1-p)^(n-k)
• Mean: E(X) = np
• Variance: V(X) = np(1-p)

Poisson Process:

• Number of rare events occurring in an interval
• Expected value: E(X) = λ

Continuous Distribution:

• Probability function: Pr(X=k) = e^(-λ) * (λ^k) / k!
• Expected value: E(X) = ∫x*f(x)dx

Mixed Distributions:

• Variance: V(X) = E(X^2) - (E(X))^2

Probability can be found with two methods:

• V(X) = E(X^2) - (E(X))^2
• F(x) = pF(x) + (1-p)F(x)
• Pr(a < X < b) = F(b) - F(a)

F(b) - F(a)mode: value that maximizes: f(X) = pf(x) + (1-p)f(x) bY Z, Pr(a < X < b) = F(x)dx

median: value that splits probability in half

Deductibles aBee I 0 if a < X < dCaps c0 if a < X < d1 ' Both Y =Y = . X-d if d < X < cd X if a < X < cIc dX - d if d < X < b Y = c-d if c < X < bc if c < X < bb1E(Y )= (x⋅d)⋅f(x)dx cExpected value = d X c Id E(Y )= (x)⋅f(x)dx + cPr(X>c)Expected value = X Change writing time :aUncovered loss = E(X) - E(Y )d 15m = 15/60 = 1/4hX = a + (b - a) · U U = x - a Change time unit change tooICONTINUOUS UNIFORM DISTRIBUTION . b - a "time unit : = 4/60 = 1/15•continuous uniform distributionstandard uniform distribution exponential distributionf (u) = 1 - eF (X) = x - af (X) = 1U XX f(x) = eFunctions :b - ab - aF (u) = uU - xF(X) = 1 - eMean : E(X) = a + bMean : E(U) = 1 median22 Mean : E(X) = 1 > m = (ln2) · 1Variance : V(U) = (b - a)Variance : V(U) = 1 1212

Variance: V(X) = 1

if A < a < b < B

Pr A < X < b = (b - a) / (2 * length of the event)

B - A = length of the domain