Unformatted text preview: cecx x ≥ where c is a positive constant. DEFINITION 9.28 Let f be a probability density function. Since we are assuming that the function F ( x ) = Z x∞ f ( t ) dt is continuous and ranges from 0 to 1 it follows by the intermediate value theorem that there is at least one number m such that F ( m ) = 1 / 2; m is called a median of the random variable X . Note that P ( X ≤ m ) = P ( X ≥ m ) = 1 / 2. EXAMPLE 9.29 The median of the uniform distribution on [ a, b ] is ( a + b ) / 2. EXAMPLE 9.30 Let f ( x ) = x < 1 ≤ x ≤ 1 / 2 1 / 2 < x < 1 1 1 ≤ x ≤ 3 / 2 3 / 2 < x . Any value in the interval [1 / 2 , 1 qualiﬁes as a median, although convention would usually take ‘the’ median to be 3 / 4....
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 Spring '07
 JonathanRogawski
 Math, Calculus, Normal Distribution, Probability, Probability theory, .........

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