SOA Probability Exam

Binomial mean

$$np$$

Binomial MGF

$$(1-p+pe^t)^n$$

Binomial PMF

$$(nCx)\cdot(p^{})^{x}\cdot(1-p)^{(n-x)}$$

Binomial Variance

$$np(1-p)$$

Discrete Uniform mean

$$\frac{(N+1)}{2}$$

Discrete Uniform MGF

$$\frac{1}{n}\cdot\frac{e^{t}-e^{t\left(n+1\right)}}{1-e^{t}}$$

Discrete Uniform PMF

$$\frac{1}{N}$$

Discrete Uniform variance

$$\frac{(n^2-1)}{12}$$

Geometric distribution does what?

number of Bernoulli trials until the first success

Geometric $E[X^2]$

$$\frac{2-p}{p^2}$$

Geometric mean

including 1st success: $\frac{1}{p}$

Geometric mean

before 1st success: $\frac{1-p}{p}$

Geometric MGF

$$\frac{pe^{t}}{1-\left(1-p)e^{t}\right.}$$

Geometric PMF

$$(1-p)^{(x-1)}\cdot p$$

$$\Sigma_0^{m}ar^{x}$$

$$\frac{a(1-r^{m})}{1-r}$$

Geometric Variance

$$\frac{(1-p)}{p^2}$$

Hypergeometric does what?

finds the probability of choosing an amount from a distinct subset

Hypergeometric Mean

$$n\cdot\frac{M}{N}$$

Hypergeometric PMF

$$\frac{(MCx)\cdot(N-MCn-x)}{NCn}$$

Hypergeometric Variance

$$n\cdot\frac{M}{N}\cdot\frac{N-M}{N}\cdot\frac{N-n}{N-1}$$

Infinite Geometric Series

$$\frac{a}{1-r}$$

multinomial $Cov[x_{i},x_{j}]$

$$-n\cdot p_{i}\cdot p_{j}$$

Multinomial mean

$$np_{i}$$

Multinomial PMF

$$\frac{n!}{k_1!\cdot k_2!\cdot...\cdot k_{r}!}\cdot p_1^{k_1}\cdot p_2^{k_2}\cdot...\cdot p_{r}^{k_{r}}$$

Multinomial variance

$$np_{i}(1-p_{i})$$

Negative Binomial does what?

of Bernoulli trials (success or failure) until the rth success (including rth success or before rth success are different equations)

Negative Binomial mean

$$\frac{r}{p}$$

Negative Binomial MGF

$$\frac{(pe^{t})^{r}}{(1-(1-p)e^{t})^{r}}$$

Negative Binomial PMF

(Including rth success)

$$(x-1Cr-1)\cdot p^{r}\cdot(1-p)^{x-r}$$

Negative Binomial PMF

(Before the rth success)

$$(r+x-1Cx)\cdot p^{r}(1-p)^{x}$$

Negative Binomial variance

$$\frac{r(1-p)}{p^2}$$

Poisson does what?

Models the # of times a random or sporadic phenomenon occurs over a period.

Poisson mean

$$\lambda$$

Poisson MGF

$$e^{\lambda\left(e^{t}-1\right)}$$

Poisson PMF

$$\frac{(e^{-\lambda})(\lambda^{x})}{x!}$$

Poisson variance

$$\lambda$$

$$\Sigma_0^{\infty}\left(x^2\cdot r^{x}\right)$$

$$\frac{r(r+1)}{\left(1-r\right)^3}$$

$$\Sigma_0^{\infty}\left(x\cdot r^{x}\right)$$

$$\frac{r}{\left(1-r\right)^2}$$

$$\left(A\cup B\right)^{\prime}$$

$$A^{\prime}\cap B^{\prime}$$

$$\left(A\cap B\right)^{\prime}$$

$$A^{\prime}\cup B^{\prime}$$

$$P\left(A\vert B\right)$$

$$\frac{P\left(A\cap B\right)}{P\left(B\right)}$$

$$\frac{P\left(A\cap B\right)}{P\left(B\right)}$$

$$\frac{P\left(B\vert A\right)\cdot P\left(A\right)}{P\left(B\vert A\right)\cdot P\left(A\right)+P\left(B\vert A^{\prime}\right)\cdot P\left(A^{\prime}\right)}$$

$$P\left(A\cap B\right)$$

$$P\left(B\vert A\right)\cdot\left(A\right)$$

$$P\left(B\vert A\right)\cdot P\left(A\right)+P\left(B\vert A^{\prime}\right)\cdot P\left(A^{\prime}\right)$$

$$P\left(B\right)$$

$$nPk$$

$$\frac{n!}{\left(n-k\right)!}$$

$$nCk$$

$$\frac{n!}{k!\left(n-k\right)!}$$

Combinations

no replacement

order does NOT matter

Permutations

no replacement

order matters

Exponential $\rightarrow$ Poisson

$$\frac{1}{\theta}=\lambda$$

Binomial $\rightarrow$ Poisson

$$np=\lambda$$

Variance : Mean

Poisson

variance = mean

Variance : Mean

Binomial

variance < mean

Variance : Mean

Negative Binomial

variance > mean

median

50th percentile

$$F_{x}\left(m\right)=.5$$

(solve for smallest m value)

Coefficient of Variation (C.V.)

$$\frac{\sigma_{x}}{\mu_{x}}$$

$$Cov\left\lbrack X,Y\right\rbrack$$

$$E\left\lbrack X\cdot Y\rbrack\right.-E\left\lbrack X\right\rbrack\cdot E\left\lbrack Y\right\rbrack$$

$$V\left(aX-bY\right)$$

(Dependent)

$$a^2\cdot V\left(X\right)+b^2\cdot V\left(Y\right)-2\cdot a\cdot b\cdot Cov\left\lbrack X,Y\rbrack\right.$$