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. CONTINUOUSTYPE 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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 Spring '08
 Zahrn
 The Land

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