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Summary: Some Useful Probabilistic Facts
For a random variable we denote the expected value of by Exp[].
Linearity of Expectation
For any two random variables 1 and 2, Exp[1 + 2] = Exp[1] + Exp[2].
Union Bound
Let E1 and E2 be two events over the same probability space. Then Pr[E1 E2] Pr[E1]+Pr[E2].
Markov's Inequality
This is the most basic inequality on the deviation of a random variable from its expectation: it
assumes very little (but is also weaker than the others that follow below). Let be a non-negative
random variable. Then for any positive k,
Pr[ k · Exp[]]
1
k
Chebishev's Inequality
Recall the definition of the variance of a random variable :
Var[]
def
= Exp[( - Exp[])2
] = Exp[2
] - (Exp[])2
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