P Formula Sheet
General Probability
Basic Probability Relationships:
Pr(A \cup B) = Pr(A) + Pr(B) - Pr(A \cap B)
Pr(A \cup B \cup C) = Pr(A) + Pr(B) + Pr(C) - Pr(A \cap B) - Pr(B \cap C) - Pr(A \cap C) + Pr(A \cap B \cap C)
Pr(A^c) = 1 - Pr(A)
Law of Total Probability:
Pr(B) = Pr(B \cap A)
De Morganβs Law:
Pr[(A \cup B)^c] = Pr(A^c \cap B^c)
Pr[(A \cap B)^c] = Pr(A^c \cup B^c)
Conditional Probability:
Pr(A|B) = \frac{Pr(A \cap B)}{Pr(B)}
Independence:
Pr(A \cap B) = Pr(A) \cdot Pr(B)
Pr(A|B) = Pr(A)
Bayesβ Theorem:
Pr(π΄π΄&|π΅π΅) =
Pr(π΅π΅|π΄π΄& ) β Pr(π΄π΄&)
β Pr(π΅π΅|π΄π΄") β Pr(π΄π΄" )#
Combinatorics:
n! = n \cdot (n-1) \cdot β¦ \cdot 2 \cdot 1
nPk = \frac{n!}{(n-k)!}
nCk = \binom{n}{k} = \binom{n}{n-k} = \frac{n!}{(n-k)! \cdot k!}
Partition = \frac{n!}{k1! \cdot k2! \cdot β¦ \cdot km!}, where k1 + k2 + β¦ + km = n
Probability = \frac{\text{number of outcomes that satisfy the event}}{\text{total number of outcomes}}
Univariate Probability Distributions
Learn both discrete and continuous cases.
Probability Mass Function (PMF):
\sum p_X(x) = 1
Pr(X = a) = 0 (continuous)
Cumulative Distribution Function (CDF):
FX(x) = Pr(X \le x) = \sum pX(i)
Pr(a < X \le b) = FX(b) - FX(a)
fX(x) = \frac{d}{dx} FX(x) (continuous)
Survival Function:
SX(x) = 1 - FX(x) = Pr(X > x)
Expected Value:
E[g(X)] = \int g(x) \cdot f_X(x) dx
E[g(X)] = \int0^\infty g'(x) \cdot SX(x) dx, for x \ge 0 and g(0) = 0
E[g(X)|j \le X \le k] = \frac{\intj^k g(x) \cdot fX(x) dx}{Pr(j \le X \le k)}
E[c] = c
E[c \cdot g(X)] = c \cdot E[g(X)]
E[g1(X) + \cdots + gn(X)] = E[g1(X)] + \cdots + E[gn(X)]
Variance, Standard Deviation, and Coefficient of Variation:
Var[X] = E[(X - \mu)^2] = E[X^2] - (E[X])^2
Var[aX + b] = a^2 \cdot Var[X]
Var[c] = 0
SD[X] = \sqrt{Var[X]}
CV[X] = \frac{SD[X]}{E[X]}
Percentiles:
The 100p^{th} percentile is the smallest value of \pip where FX(\pi_p) \ge p.
Modes:
The mode(s) of a random variable is/are the value(s) where the probability function is maximized.
Discrete Distributions
Discrete Uniform:
PMF: \frac{1}{b - a + 1}, x = a, a+1, β¦, b
Mean: \frac{a+b}{2}
Variance: \frac{(b-a+1)^2 - 1}{12}
Binomial:
PMF: \binom{n}{x} p^x (1-p)^{n-x}, x = 0, 1, β¦, n
Mean: np
Variance: np(1-p)
Special Properties: Sum of independent binomials with same p is binomial (n = \sum n_i, p)
Hypergeometric:
PMF: \frac{\binom{K}{x} \cdot \binom{N-K}{n-x}}{\binom{N}{n}}
Mean: n \cdot \frac{K}{N}
Variance: n \cdot \frac{K}{N} \cdot \frac{N-K}{N} \cdot \frac{N-n}{N-1}
Geometric (X = # trials until 1st success):
PMF: (1-p)^{x-1}p, x = 1, 2, 3, β¦
Mean: \frac{1}{p}
Variance: \frac{1-p}{p^2}
Memoryless property: (X-c | X > c) \sim X
Geometric (X = # failures before 1st success):
PMF: (1-p)^x p, x = 0, 1, 2, β¦
Mean: \frac{1-p}{p}
Negative Binomial (X = # trials until rth success):
PMF: \binom{x-1}{r-1} p^r (1-p)^{x-r}, x = r, r+1, r+2, β¦
Mean: \frac{r}{p}
Variance: \frac{r(1-p)}{p^2}
Special Properties: Sum of r independent geometric(p) is negative binomial (r = \sum r_i, p)
Negative Binomial (X = # failures until rth success):
PMF: \binom{r+x-1}{r-1} p^r (1-p)^x, x = 0, 1, 2, β¦
Mean: \frac{r(1-p)}{p}
Poisson:
PMF: \frac{e^{-\lambda} \cdot \lambda^x}{x!}, x = 0, 1, 2, β¦
Mean: \lambda
Variance: \lambda
Special Properties:
Sum of independent Poissons is Poisson (\lambda = \sum \lambda_i)
Non-overlapping intervals are independent
Continuous Distributions
Continuous Uniform:
PDF: \frac{1}{b-a}, a \le x \le b
CDF: \frac{x-a}{b-a}
Mean: \frac{a+b}{2}
Variance: \frac{(b-a)^2}{12}
(X|c < X < d) \sim Uniform(c, d)
(X-c | X > c) \sim Uniform(0, b-c)
Exponential:
PDF: \frac{1}{\theta} e^{-\frac{x}{\theta}}, x > 0
CDF: 1 - e^{-\frac{x}{\theta}}
Mean: \theta
Variance: \theta^2
Memoryless property: (X-c | X > c) \sim X
Gamma:
PDF: \frac{x^{\alpha-1} e^{-\frac{x}{\theta}}}{\Gamma(\alpha) \cdot \theta^\alpha}, x > 0
CDF: 1 - \sum_{k=0}^{\alpha-1} \frac{e^{-\frac{x}{\theta}} (\frac{x}{\theta})^k}{k!}, \alpha = 1, 2, 3, β¦
Mean: \alpha \theta
Variance: \alpha \theta^2
Special properties: Sum of \alpha independent exponentials(\theta) is Gamma(\alpha, \theta)
Normal:
PDF: \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} (\frac{x-\mu}{\sigma})^2}, - \infty < x < \infty
Z = \frac{X - \mu}{\sigma}
CDF: Pr(Z \le z) = \Phi(z)
Mean: \mu
Variance: \sigma^2
Symmetry: Pr(Z \le z) = Pr(Z \ge -z)
Special properties:
Sum of independent normals is normal (\mu = \sum \mui, \sigma^2 = \sum \sigmai^2)
LogNormal:
PDF: \frac{1}{x \sigma \sqrt{2\pi}} e^{-\frac{1}{2} (\frac{ln x - \mu}{\sigma})^2}, x > 0
CDF: \Phi(\frac{ln x - \mu}{\sigma})
Mean: e^{\mu + \frac{1}{2} \sigma^2}
Variance: e^{2\mu + \sigma^2} (e^{\sigma^2} - 1)
If X is lognormal then lnX is normal
Product of independent lognormals is lognormal (\mu = \sum \mui, \sigma^2 = \sum \sigmai^2)
Beta:
PDF: \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} x^{a-1} (1-x)^{b-1}, 0 \le x \le 1
CDF: No closed form
Mean: \frac{a}{a+b}
Variance: \frac{ab}{(a+b)^2 (a+b+1)}
Beta(1, 1) is Uniform(0, 1)
Multivariate Probability Distributions
Joint PMF and CDF:
\sumx \sumy p_{X,Y}(x, y) = 1
F{X,Y}(x, y) = \sum{i \le x} \sum{j \le y} p{X,Y}(i, j)
F{X,Y}(x, \infty) = FX(x)
F{X,Y}(\infty, y) = FY(y)
S_{X,Y}(x, y) = Pr[(X > x) \cap (Y > y)]
Marginal Distributions and Conditional Distributions:
pX(x) = \sumy p_{X,Y}(x, y)
pY(y) = \sumx p_{X,Y}(x, y)
p{X|Y}(x|Y=y) = \frac{p{X,Y}(x, y)}{p_Y(y)}
Joint Expected Value and Conditional Expectation:
E[g(X, Y)] = \sumx \sumy g(x, y) \cdot p_{X,Y}(x, y)
E[X|Y=y] = \sumx x \cdot p{X|Y}(x|Y=y)
Weighted Average:
For conditional random variables of Y denoted as C1 and C2 with weights a1 + a2 = 1
fY(y) = a1 f{C1}(y) + a2 f{C_2}(y)
FY(y) = a1 F{C1}(y) + a2 F{C_2}(y)
SY(y) = a1 S{C1}(y) + a2 S{C_2}(y)
E[Y^k] = a1 E[C1^k] + a2 E[C2^k]
Double Expectation and Law of Total Variance:
E[X] = E[E[X|Y]]
Var[X] = E[Var[X|Y]] + Var[E[X|Y]]
Covariance and Correlation Coefficient:
Cov[X, Y] = E[XY] - E[X]E[Y]
Cov[aX, bY] = ab \cdot Cov[X, Y]
Cov[X, X] = Var[X]
Var[aX + bY] = a^2 Var[X] + b^2 Var[Y] + 2ab \cdot Cov[X, Y]
\rho_{X,Y} = Corr[X, Y] = \frac{Cov[X, Y]}{\sqrt{Var[X]Var[Y]}}
Independence:
F{X,Y}(x, y) = FX(x) \cdot F_Y(y)
f{X,Y}(x, y) = fX(x) \cdot f_Y(y)
f{X|Y}(x|y) = fX(x)
f{Y|X}(y|x) = fY(y)
E[g(X) \cdot h(Y)] = E[g(X)] \cdot E[h(Y)]
Cov[X, Y] = 0
\rho_{X,Y} = 0
Multinomial Distribution:
Pr(X1 = x1, β¦, Xk = xk) = \frac{n!}{x1! \cdot β¦ \cdot xk!} \cdot p1^{x1} \cdot β¦ \cdot pk^{xk}
E[Xi] = npi
Var[Xi] = npi(1 - p_i)
Cov[Xi, Xj] = -npi pj, for i \ne j
Expectation and Variance for Sum and Average of IID Random Variables:
S = X1 + β¦ + Xn
\overline{X} = [X1 + β¦ + Xn]/n
E[S] = n \cdot E[X_i]
E[\overline{X}] = E[X_i]
Var[S] = n \cdot Var[X_i]
Var[\overline{X}] = (1/n) \cdot Var[X_i]
Central Limit Theorem:
The sum or average of a large number of independent and identically distributed (i.i.d.) random variables approximately follows a normal distribution.
Order Statistics:
X_{(k)} = kth order statistic
X{(1)} = min(X1, X2, β¦, Xn)
X{(n)} = max(X1, X2, β¦, Xn)
For i.i.d. random variables,
S{X{(1)}}(x) = [S_X(x)]^n
F{X{(n)}}(x) = [F_X(x)]^n
f{X{(k)}}(x) = \frac{n!}{(k-1)! (n-k)!} \cdot [FX(x)]^{k-1} \cdot fX(x) \cdot [S_X(x)]^{n-k}
Insurance and Risk Management
Learn both discrete and continuous cases
Unreimbursed Loss, L
If X is the loss and Y is the payment (ie reimbursed loss), then X = Y + L
L = X - Y, and E[L] = E[X] - E[Y]
Deductible
Y = 0, X <= d , X-d, X > d
E[Y] = integral(x-d) * fX(x) dx from d to infinity = integral(SX(x)) dx from d to infinity
For exponential: d * Pr(X > d)
Policy Limit
Y = X, X <= u, u, X > u
E[Y] = integral(x * fX(x) dx) from 0 to u + u * SX(u) = integral(SX(x) dx) from 0 to u
For exponential: u * Pr(X < u)
Deductible and Policy Limit
Y= 0, X <= d, X-d, d < X < d+u, u, X >= d+u
E[Y] = integral((x-d) * fX(x) dx) from d to d+u + u * SX(d+u) = integral(SX(x) dx) from d to d+u
For exponential: theta * Pr( d < X < d+u)