Probability

-$1/n$
-$(a+b)/2$
-$((b-a+1)^2) - 1/12$

probabilities must satisfy additive property (sum=1)

Let Y be the # of successes in the n attempts
$\binom{n}{y} p^y * q^(n-y)$
$n*p$
$n*p*q$

E[X] = $1/p$, but if N=0 $q/p$
Var[X] = $q/(p^2)$

Memoryless & Mode = 1

x = trial where the r^th success occurs
Pr(X=x): $\binom{x-1}{r-1}p^r(1-p)^{x-r}, \quad x=r,r+1,r+2,\dots$
E[X] = $(rq)/p$
Var[X] = $rq/(p^2)$

(ORDER MATTERS!!!) Given set of n distinguishable objects, an ordered selection of r different elements is chosen from the n objects
Denoted by nPr = $ \frac{n!}{(n-r)!}$

Pr(E’) = 1 - $\frac{nP_r}{n^r}$

order doesn't matter!!
Given a set of n distringuishable objects, an unordered selection of r different elements of set is called combination of n objects chosen r at a time
$nCr = \frac{P}{r!} = \frac{n!}{r!(n-r)!}$

nCr = nC(n-r)

# of partitions of n distinct objects into k subsets of sizes $r_1, r_2 … r_k $ where $r_1+r_2+r_k = n$ denoted by $n \choose{r_1,r_2,r_k}$ = $ \frac{n!}{r_1!r_2!…r_k!} $

$(x+y)^n=\sum_{r=0}^{n} nC_r (x^r)(y^{n-r})$

$(x_1+x_2+...+x_r)^n = \sum_{n_1+...+n_r=n}^{} \frac{n!}{n_1!,n_2!...n_r!}(x_r)^{n_r}$

$Pr(E\cup F)=Pr(E)+Pr(F)-Pr(E\cap F)$

$Pr(A | B)=\frac{Pr(A\cap B)}{Pr(B)}$

$e^(-λ) * (λ^k)/k!$
E[X] and Var[X] both = λ

Supposed $Z_i$ are independent Poisson RV with mean $λ$. Then $Z = Z_1 + Z_2$ is a poisson RV with mean $E[Z] = λ_1 + λ_2$

Int floor OR the # and #-1

E[Y] = a*E[X] + b
Var[Y] = a^2 * Var[X]

$100 * (sd)/(mean)$

f(x) = $\frac{1}{b-a}$

F(x) = $\frac{x-a}{b-a}$

E[X] = $\frac{a+b}{2}$

Var[X] =$\frac{(b-a)^2}{12}$

f(x) = $\frac{1}{θ} * e^{\frac{-x}{ θ}}$

F(x) = $1 - e^{\frac{-x}{ θ}}$

E[X] = $θ$

Var[X] =$θ^2$

E[X | X>a] = θ+a
Var[X | X>a] = θ^2

E[X*Y] - E[X]E[Y]

Cov(X,Y) / SDx * SDy

f(x) = $\frac{1}{(a-1)!} * \frac{x^{a-1}}{ θ^a}*e^{-x/θ}$

Let X ~ Exp(θ) and Y ~ Gamma (a,θ)

E[Y] = a*θ
Var[Y] = a*θ^2

Γ(n)=(n−1)!

(a-1)* θ

$E[X] = k + θ*e^{-k/ θ}$

Q_3 - Q_1 = SD(π75 - π25)

$\frac{1}{1-Bt}$

x=0

= B * ln(2)

SD / sqrt(n)

E[X} and Var[X] same as binomial
Cov is $-n*p_1*p_2$