IPS

What is the P(X=x) of a Bernoulli distribution

What is the P(X=x) of a Bernoulli distribution

P(X=1) = p and P(X=0) = 1−p.

What is the E[X] of a Bernoulli Distribution

p

What is the Var(X) of a Bernoulli Distribution

p(1-p)

What is the P(X=k) of a Binomial distribution

P(X=k) = (nk)p^k(1−p)^n−k

What is the expected value (E[X]) of a Binomial distribution

np

What is the variance (Var(X)) of a Binomial distribution

np(1 − p)

What is the P(X=x) of a Geometric distribution

P(X = k) = p(1 − p)^(k−1)

What is the expected value (E[X]) of a Geometric distribution

1/p

What is the variance Var(X) of a Geometric distribution

(1 − p)/p^2.

What is the P(X=x) of a Poisson distribution

P(X=k) = (μ^k/ k!) e^−μ

What is the expected value (E[X]) of a Poisson distribution

μ

What is the variance (Var(X)) of a Poisson distribution

μ

What is the probability density function (PDF) of an Exponential distribution

f(x) = λe^(−λx)

What is the cumulative distribution function (CDF) of an Exponential distribution

F(x) = 1 − e^−λx

What is the expected value (E[X]) of an Exponential distribution

1/λ

What is the variance (Var(X)) of an Exponential distribution

1/λ^2

What is the probability density function (PDF) of a Normal distribution

f(x) = (1/σ√(2π))e^−(x−μ)^2 / (2σ^2)

What is the probability density function (PDF) of a Uniform distribution

f(x) = 1 / (b − a)

What is the cumulative distribution function (CDF) of a Uniform distribution

F(x) = (x − a) / (b − a)

What is the expected value (E[X]) of a Uniform distribution

(a + b)/2

What is the variance (Var(X)) of a Uniform distribution

(b − a)^2/12

Cov(X,Y)

E[(X−E[X])(Y −E[Y])]

Cov(X, Y) (Alternate expression)

E [XY] − E [X] E [Y]

Var(X+Y)=

Var(X)+Var(Y)+2Cov(X,Y)

Correlation coefficient P(X,Y)=

Cov(X,Y)/ sqrt(Var(X)Var(Y))

Var(Y)/a²

P(|Y−E[Y]| ≥ a) ≤

Var(X) =

E[(X-E[X])²]

Sn²=

n/n-1*(Var(X))

(AnB)u(CnD) =

(AuC)n(BuD)

To divide items amongst things

C(n+r-1)_(r-1)

Number of ways you can order n items

N!

Number of ways you can order c1,c2,….,cn objects into groups

n!c1!c2!…cn!

Orders of n objects of which n1 are indistinguishable

n1 objects of which n2 are indistinguishable

.

.

.

nr-1 of which n are indistinguishable

n!/n1!n2!…nr!

How many distinct ordered sets of length r can be formed from n distinct objects

Pn_r = n!/(n-r)!

A’nB’

(AuB)’

(AnB)’

A’uB’