STA 711 : Probability & Measure
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STA 205: Probability & Measure Theory
This is sometimes denoted simply “X−1(B) ⊂ F.” Since the probability measure P is only defined on sets F ∈ F, a random variable must satisfy this condition if we are to be able to find the probability Pr[X ∈ B] for each Borel set B, or even if we want to find the distribution function (DF) FX(b) ≡ Pr[X ≤ b] for each rational number b. Note that set-inverses are rather well-behaved functions fro...
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A sequence of elements an of R d converges to a limit a if and only if, for each ǫ > 0, the sequence an eventually lies within a ball of radius ǫ centered at a. It’s okay if the first few (or few million) terms lie outside that ball— and the number of terms that do lie outside the ball may depend on how big ǫ is (if ǫ is small enough it typically will take millions of terms before the remaining...
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