Section3

# Section3 - SECTION NOTE 3 ORIE360/560 FALL 2004 CHONG WANG...

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Unformatted text preview: SECTION NOTE 3 ORIE360/560, FALL 2004 CHONG WANG 1. Continuous Random Variables X is a continuous random variable if there exists a probability density function (pdf) of X , which satisfies the following properties. (a) f ( x ) ≥ 0. (b) R ∞-∞ f ( x ) dx = 1. (c) P ( X ∈ A ) = R A f ( x ) dx . The relationship between the cumulative distribution F ( · ) and the probability density f ( · ) is expressed by F ( a ) = P { X ∈ (-∞ ,a ] } = Z a-∞ f ( x ) dx. Differentiating both sides yields d da F ( a ) = f ( a ) . When we know one of the pdf and cdf, these relationships could be used to calculate the other. Example : (Problem 8, Chapter 4) If the density function of X equals f ( x ) = ‰ ce- 2 x < x < ∞ , x ≤ . find c . What is P ( X > 2)? Solution 1 = Z ∞-∞ f ( x ) dx = Z ∞ ce- 2 x dx = c/ 2 . c has to be 2. P ( X > 2) = Z ∞ 2 2 e- 2 x dx = e- 4 Date : September 13, 2004 - September 17, 2004 ....
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Section3 - SECTION NOTE 3 ORIE360/560 FALL 2004 CHONG WANG...

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