Isye 2027

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Unformatted text preview: ≥ 0, the k th moment is given by 1 E [X k ] = uk du = 0 and the variance is 1 3 uk+1 k+1 − ( 1 )2 = 2 1 12 . 1 = 0 1 k+1 (if U is uniformly distributed over [0, 1]), 78 3.4 CHAPTER 3. CONTINUOUS-TYPE RANDOM VARIABLES Exponential distribution A random variable T has the exponential distribution with parameter λ > 0 if its pdf is given by λe−λt t ≥ 0 0 else. fT (t) = The pdfs for the exponential distributions with parameters λ = 1, 2, and 4 are shown in Figure 3.6. Note that the initial value of the pdf is given by fT (0) = λ, and the rate of decay of fT (t) as f (t) T !=4 !=2 !=1 t Figure 3.6: The pdfs for the exponential distributions with λ = 1, 2, and 4. t increases is also λ. Of course, the area under the graph of the pdf is one for any value of λ. The mean is decreasing in λ, because the larger λ is, the more concentrated the pdf becomes towards the left. The CDF evaluated at a t ≥ 0, is given by t t fT (s)ds = FT (t) = −∞ t = 1 − e−λt . λe−λs ds = −e−λs 0 0 Therefore, in general, 1 − e−λt t ≥ 0 0...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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