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 Morgan’s 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:

  • $Pr(A \cap B) = Pr(A) \cdot Pr(B)$

  • $Pr(A|B) = Pr(A)$

• Bayes’ Theorem:

  • Pr(𝐴𝐴&|𝐵𝐵) =

    Pr(𝐵𝐵|𝐴𝐴& ) ⋅ Pr(𝐴𝐴&)

    ∑ Pr(𝐵𝐵|𝐴𝐴") ⋅ Pr(𝐴𝐴" )#

• Combinatorics:

  • $n! = n \cdot (n-1) \cdot … \cdot 2 \cdot 1$

  • $<em>nP</em>k = \frac{n!}{(n-k)!}$

  • $<em>nC</em>k = \binom{n}{k} = \binom{n}{n-k} = \frac{n!}{(n-k)! \cdot k!}$

  • Partition = $\frac{n!}{k<em>1! \cdot k</em>2! \cdot … \cdot k<em>m!}$, where $k</em>1 + k<em>2 + … + k</em>m = n$

  • 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):

  • $F<em>X(x) = Pr(X \le x) = \sum p</em>X(i)$

  • $Pr(a < X \le b) = F<em>X(b) - F</em>X(a)$

  • $f<em>X(x) = \frac{d}{dx} F</em>X(x)$ (continuous)

• Survival Function:

  • $S<em>X(x) = 1 - F</em>X(x) = Pr(X > x)$

• Expected Value:

  • $E[g(X)] = \int g(x) \cdot f_X(x) dx$

  • $E[g(X)] = \int<em>0^\infty g'(x) \cdot S</em>X(x) dx$, for $x \ge 0$ and $g(0) = 0$

  • $E[g(X)|j \le X \le k] = \frac{\int<em>j^k g(x) \cdot f</em>X(x) dx}{Pr(j \le X \le k)}$

  • $E[c] = c$

  • $E[c \cdot g(X)] = c \cdot E[g(X)]$

  • $E[g<em>1(X) + \cdots + g</em>n(X)] = E[g<em>1(X)] + \cdots + E[g</em>n(X)]$

• 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 $\pi<em>p$ where $F</em>X(\pi_p) \ge p$.

• Modes:

  • The mode(s) of a random variable is/are the value(s) where the probability function is maximized.

Discrete Distributions

• Discrete Uniform:

  • PMF: $\frac{1}{b - a + 1}$, $x = a, a+1, …, b$

  • Mean: $\frac{a+b}{2}$

  • Variance: $\frac{(b-a+1)^2 - 1}{12}$

• Binomial:

  • PMF: $\binom{n}{x} p^x (1-p)^{n-x}$, $x = 0, 1, …, n$

  • 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: $(1-p)^{x-1}p$, $x = 1, 2, 3, …$

  • 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: $(1-p)^x p$, $x = 0, 1, 2, …$

  • Mean: $\frac{1-p}{p}$

• Negative Binomial (X = # trials until rth success):

  • PMF: $\binom{x-1}{r-1} p^r (1-p)^{x-r}$, $x = r, r+1, r+2, …$

  • 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: $\binom{r+x-1}{r-1} p^r (1-p)^x$, $x = 0, 1, 2, …$

  • Mean: $\frac{r(1-p)}{p}$

• Poisson:

  • PMF: $\frac{e^{-\lambda} \cdot \lambda^x}{x!}$, $x = 0, 1, 2, …$

  • 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: $1 - \sum_{k=0}^{\alpha-1} \frac{e^{-\frac{x}{\theta}} (\frac{x}{\theta})^k}{k!}$, $\alpha = 1, 2, 3, …$

  • 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 ($\mu = \sum \mu<em>i, \sigma^2 = \sum \sigma</em>i^2$)

• 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 ($\mu = \sum \mu<em>i, \sigma^2 = \sum \sigma</em>i^2$)

• 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:

  • $\sum<em>x \sum</em>y p_{X,Y}(x, y) = 1$

  • $F<em>{X,Y}(x, y) = \sum</em>{i \le x} \sum<em>{j \le y} p</em>{X,Y}(i, j)$

  • $F<em>{X,Y}(x, \infty) = F</em>X(x)$

  • $F<em>{X,Y}(\infty, y) = F</em>Y(y)$

  • $S_{X,Y}(x, y) = Pr[(X > x) \cap (Y > y)]$

• Marginal Distributions and Conditional Distributions:

  • $p<em>X(x) = \sum</em>y p_{X,Y}(x, y)$

  • $p<em>Y(y) = \sum</em>x p_{X,Y}(x, y)$

  • $p<em>{X|Y}(x|Y=y) = \frac{p</em>{X,Y}(x, y)}{p_Y(y)}$

• Joint Expected Value and Conditional Expectation:

  • $E[g(X, Y)] = \sum<em>x \sum</em>y g(x, y) \cdot p_{X,Y}(x, y)$

  • $E[X|Y=y] = \sum<em>x x \cdot p</em>{X|Y}(x|Y=y)$

• Weighted Average:

  • For conditional random variables of Y denoted as $C<em>1$ and $C</em>2$ with weights $a<em>1 + a</em>2 = 1$

    • $f<em>Y(y) = a</em>1 f<em>{C</em>1}(y) + a<em>2 f</em>{C_2}(y)$

    • $F<em>Y(y) = a</em>1 F<em>{C</em>1}(y) + a<em>2 F</em>{C_2}(y)$

    • $S<em>Y(y) = a</em>1 S<em>{C</em>1}(y) + a<em>2 S</em>{C_2}(y)$

    • $E[Y^k] = a<em>1 E[C</em>1^k] + a<em>2 E[C</em>2^k]$

• 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:

  • $F<em>{X,Y}(x, y) = F</em>X(x) \cdot F_Y(y)$

  • $f<em>{X,Y}(x, y) = f</em>X(x) \cdot f_Y(y)$

  • $f<em>{X|Y}(x|y) = f</em>X(x)$

  • $f<em>{Y|X}(y|x) = f</em>Y(y)$

  • $E[g(X) \cdot h(Y)] = E[g(X)] \cdot E[h(Y)]$

  • $Cov[X, Y] = 0$

  • $\rho_{X,Y} = 0$

• Multinomial Distribution:

  • $Pr(X<em>1 = x</em>1, …, X<em>k = x</em>k) = \frac{n!}{x<em>1! \cdot … \cdot x</em>k!} \cdot p<em>1^{x</em>1} \cdot … \cdot p<em>k^{x</em>k}$

  • $E[X<em>i] = np</em>i$

  • $Var[X<em>i] = np</em>i(1 - p_i)$

  • $Cov[X<em>i, X</em>j] = -np<em>i p</em>j$, for $i \ne j$

• Expectation and Variance for Sum and Average of IID Random Variables:

  • $S = X<em>1 + … + X</em>n$

  • $\overline{X} = [X<em>1 + … + X</em>n]/n$

  • $E[S] = n \cdot E[X_i]$

  • $E[\overline{X}] = E[X_i]$

  • $Var[S] = n \cdot Var[X_i]$

  • $Var[\overline{X}] = (1/n) \cdot Var[X_i]$

• 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

  • $X<em>{(1)}$ = min($X</em>1, X<em>2, …, X</em>n$)

  • $X<em>{(n)}$ = max($X</em>1, X<em>2, …, X</em>n$)

  • For i.i.d. random variables,

    • $S<em>{X</em>{(1)}}(x) = [S_X(x)]^n$

    • $F<em>{X</em>{(n)}}(x) = [F_X(x)]^n$

    • $f<em>{X</em>{(k)}}(x) = \frac{n!}{(k-1)! (n-k)!} \cdot [F<em>X(x)]^{k-1} \cdot f</em>X(x) \cdot [S_X(x)]^{n-k}$

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)