Isye 2027

# Determine whether y is a continuous type or discrete

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Unformatted text preview: nd the approximate numerical value of P {|X − 50| ≥ 6}. (c) Find the Chebychev upper bound on P {|X − 50| ≥ 6}. Compare it to your answer to part (b). Solution: (a) The random variable X has the binomial distribution with parameters n = 60 and p = 5/6. It thus has mean µX = np = 50 and standard deviation σX = np(1 − p) ≈ 2.887. Thus, 2 Use of the continuity correction is speciﬁed here to be deﬁnite, although n is so large that the continuity correction is not very important. 96 CHAPTER 3. CONTINUOUS-TYPE RANDOM VARIABLES X −50 2.89 is approximately a standard normal random variable. Therefore, P {X ≤ 52} = P {X ≤ 52.5} = P =P X − 50 ≤ 0.866 2.887 X − 50 52.5 − 50 ≤ 2.887 2.887 ≈ Φ(0.866) ≈ 0.807. The Gaussian approximation in this case is fairly accurate: numerical calculation, using the binomial distribution, yields that P {X ≤ 52} ≈ 0.8042. (b) P {|X − 50| ≥ 6} = P {X ≤ 44 or X ≥ 56} = P {X ≤ 44.5 or X ≥ 55.5} X − 50 X − 50 =P ≤ −1.905 or ≥ 1.905 2.887 2.887 ≈ 2Q(1....
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