Random Variable Formulas

Standard deviation of a discrete random variable

θ_x= sqrt(E(x_i-M_x)²P_i))

Mean of a discrete random variable

M_x=E(x_iP_i)

Mean of a sum of random variables

M_x+-y=M_x+-M_y

Mean of a linear combination of random variables

M_ax+by=aM_x+bM_y

Standard deviation of a linear combination of random variables

θ_ax+by= sqrt(a²θ_x²+b²θ_y²) if x and y are independent

Standard deviation of the sum of 2 independent random variables

θ_s=sqrt(θ_x²+θ_y²)

Standard deviation of the difference of the independent random variables

θ_D=sqrt(θ_x²+θ_y²)

Standard deviation of a binomial random variable

θ_x=sqrt(np(1-p))

Mean of a binomial random variable

M_x=np

Geometric probability formula

P(X=x) = (1-p)^x-1p

geometric distribution mean

M_x=1/p

geometric distribution standard deviation

θ=(sqrt(1-p))/p

Binomial probability

P(X=x) = (n/x)p^x(1-p)^n-x

Binomial coefficient

(n/x)= n!/x!(n-x)!

Linear transformation of mean

M_a+bx=a + bM_x

Linear Transformation of standard deviation

θ_a+bx= lblθ_x

Binomial setting

BINS

BINS

Binary, Independent, Number of trials, Same probability

Geometric setting

BITS

BITS

Binary, Independent, Trials until success, Same probability