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Unformatted text preview: F (u) X 1.0 0.75 0.5 . 0.25 0 u 5 10 15 Figure 3.3: An example of a CDF. 72 CHAPTER 3. CONTINUOUS-TYPE RANDOM VARIABLES Solution: (a) P {X ≤ 1} = FX (1) = 0.05. (b) P {X ≤ 10} = FX (10) = 0.75. (c) P {X ≥ 10} = 1 − P {X < 10} = 1 − FX (10−) = 0.5. (d) P {X = 10} = FX (10) = 0.25. (e) P {|X − 5| ≤ 0.1} = P {4.9 ≤ X ≤ 5.1} = P {X ≤ 5.1} − P {X < 4.9} = FX (5.1) − FX (4.9−) = 0.5 − 0.245 = 0.255. The following proposition follows from the axioms of probability and the definition of CDFs. The proof is omitted, but a proof of the only if part can be given along the lines of the proof of Proposition 3.1.2. Proposition 3.1.5 A function F is the CDF of some random variable if and only if it has the following three properties: F.1 F is nondecreasing F.2 limc→+∞ F (c) = 1 and limc→−∞ F (c) = 0 F.3 F is right continuous (i.e. FX (c) = FX (c+) for all c ). Example 3.1.6 Which of the six functions shown in Figure 3.4 are valid CDFs? For each one that is not valid, state a property from Proposition 3.1.5 that...
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