Independence of X and Y means that for all i and j:

Expected value E[X]:

Linearity of expectation (no independence needed):

Product of expectations of independent R.V.s

Variance and standard deviation

Harmonic numbers

Approximations to e:

Binomial distribution with parameters n and p:

Mean = np, Variance = np(1-p)

Poisson distribution with parameter l :

Mean = l , Variance = l

n balls into n bins Pr[bin i has k balls] » e^-1/k! (Poisson with parameter 1)

Random permutation of n elements, Pr[k fixed points] » e^-1/k! (Poisson with parameter 1)

Birthday Paradox, m balls into n bins

Pr[all bins have < 2 balls] <= e^-m(m-1)/2n

Markov:

Chebyshev:

Coupon collecting, for high probability of collecting all coupons:

Geometric random variable X:

Chernoff bounds for sum of indicator variables from Poisson trials. Upper tail:

Upper tail, when d < 2e –1

Upper tail when d > 2e-1

Lower tail:

Inclusion-exclusion. If T_k is any outcome where exactly k of the properties E₁…E_n hold, then the general formula is