121616949-math.229

121616949-math.229 - called the mean is generally most...

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9.8 Probability 215 An even function is one that is symmetric around the y axis. EXAMPLE 9.31 If f is an even probability density function then one of the medians is 0. If f ( x ) > 0 on [ - δ, δ ] for some δ > 0, then 0 is the only median. In particular, the median of the standard normal distribution is 0. DEFINITION 9.32 If a probability density function f has a global maximum at x = c then c is a mode of the random variable X . There need not be a single mode; for example, in the uniform distribution every number between a and b is a mode. The mode may not even exist, though it is a bit tricky to come up with an example. EXAMPLE 9.33 It is somewhat difficult to devise a continuous probability density function for which there is no mode, but easy if we give up continuity. Consider f ( x ) = 0 x < - 1 x + 1 - 1 x < 0 0 x = 0 - x + 1 0 < x 1 0 1 < x . Sketch this graph; it is then apparent that f has no global maximum and hence no mode. In practice, both the mode and the median are useful, but the expected value, also
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Unformatted text preview: called the mean, is generally most useful. Following our discussion of discrete probability, the deﬁnition should not be surprising. DEFINITION 9.34 The mean of a continuous random variable X with probability density function f is μ = E ( X ) = Z ∞-∞ xf ( x ) dx , provided the integral converges. When the mean exists it is unique, since it is the result of an explicit calculation. The mean does not always exist. Let us look more closely at the deﬁnition of the mean. It is in fact essentially identical to the deﬁnition of the center of mass of a one-dimensional beam. The probability density function f plays the role of the physical density function, but now the “beam” has inﬁnite length. If we consider only a ﬁnite portion of the beam, say between a and b , then the...
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