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Unformatted text preview: ve integers. For example, P {X ≤ 2.5} = 0.6778 and P {X ≤ 2.5} = 0.6537. 94 CHAPTER 3. CONTINUOUSTYPE RANDOM VARIABLES Since X is an integervalued random variable, P {X ≤ 2.5} = P {X ≤ 2}. Therefore, we have:
P {X ≤ 2} = P {X ≤ 2.5} ≈ P {X ≤ 2.5}.
So, P {X ≤ 2.5} is a fairly good approximation to P {X ≤ 2}. In particular, it is a better approximation than P {X ≤ 2} is–in this case P {X ≤ 2} = 0.5. In general, when X is an integervalued
random variable, the Gaussian approximation with the continuity correction is:
P {X ≤ k } ≈ P {X ≤ k + 0.5}
P {X ≥ k } ≈ P {X ≥ k − 0.5}. (Gaussian approximation with continuity correction) The example just given shows that, with the continuity correction, the Gaussian approximation
is fairly accurate even for small values of n. The approximation improves as n increases. The CDF
of X and X are shown in Figure 3.12 in case X has the binomial distribution with parameters
n = 30 and p = 0.2. Figure 3.12: The CDF of a...
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 Spring '08
 Zahrn
 The Land

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