math319_S10_cdf_properties

math319_S10_cdf_properties - F b = 0 since b n n ≥ 1 was...

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Probability and Statistics – Math319, Spring 2010 Properties of the Cumulative Distribution Function April 15, 2010 Let X be a random variable with cumulative distribution function F . 1. F is a non-decreasing function. Proof. Let a,b R where a b . Thus F ( a ) F ( b ). That is, F is a non-decreasing function. 2. lim b →∞ F ( b ) = 1. Proof. Let { b n } n 1 be a sequence of values increasing to . Thus lim n →∞ F ( b n ) = 1. So, lim b →∞ F ( b ) = 1 since { b n } n 1 was an arbitrary sequence of values increasing to . 3. lim b →-∞ F ( b ) = 0. Proof. Let { b n } n 1 be a sequence of values decreasing to -∞ . Thus lim n →∞ F ( b n ) = 0. So, lim b →-∞
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Unformatted text preview: F ( b ) = 0 since { b n } n ≥ 1 was an arbitrary sequence of values decreasing to-∞ . 4. F is right continuous. Proof. Let b ∈ R and { b n } n ≥ 1 be a decreasing sequence of values that converges to b . Thus lim n →∞ F ( b n ) = F ( b ). So, F is right continuous. Notes: • Let a,b ∈ R where a < b . We can write { X ≤ b } as { X ≤ b } = { X ≤ a } ∪ { a < X ≤ b } . Thus P ( a < X ≤ b ) = F ( b )-F ( a ), for all a < b . • P ( X < b ) = = = lim n →∞ F ± b-1 n ² ....
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math319_S10_cdf_properties - F b = 0 since b n n ≥ 1 was...

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