NPRE 475 HW 7

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University of Illinois, Urbana Champaign **We aren't endorsed by this school
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NPRE 475
Subject
Statistics
Date
Aug 28, 2023
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3
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Name: Adedamola Oladunni NPRE 475 HW 7 1. The probability density function (pdf) of the two parameter Weibull distribution used in modelling wind duration curves is: ࠵?(࠵?) = ࠵? ࠵? ) ࠵? ࠵? + ࠵?"࠵? ࠵? "% ࠵? ࠵? ( ࠵? Where: k = shape parameter or slope C = scale parameter or characteristic wind speed As special cases, deduce the forms of: 1. The Rayleigh distribution, For Rayleigh distribution, k=2 Substituting k=2 in the W(v) equation ࠵?࠵?࠵? = 2 ࠵? ) ࠵? ࠵? + )"* ࠵? "% + , ( " ࠵?࠵?࠵? = 8 ࠵?࠵? ࠵? ࠵? : ࠵? . "% ࠵? ࠵? ( ࠵? / Substituting k=2 into the ccdf equation ࠵?࠵?࠵?࠵? = ࠵? "% + , ( $ ࠵?࠵?࠵?࠵? = ࠵? "% ࠵? ࠵? ( ࠵? 2. The Exponential distribution For Exponential distribution, k=1 Substituting k=1 into the W(v) equation ࠵?࠵?࠵? = 2 ࠵? ) ࠵? ࠵? + *"* ࠵? "% + , ( % ࠵?࠵?࠵? = 8 ࠵? ࠵? : ࠵? "% ࠵? ࠵? ( Substituting k=1 into the ccdf equation ࠵?࠵?࠵?࠵? = ࠵? "% + , ( $ ࠵?࠵?࠵?࠵? = ࠵? "% ࠵? ࠵? ( 2. Consider the exponential probability density function (pdf): ࠵?(࠵?)࠵?࠵? = ࠵? ࠵? ࠵? " ࠵? ࠵? ࠵?࠵? 1. Apply the normalization condition to prove that it is indeed a probability density function (pdf). For normalization, ࠵?(࠵?)࠵?࠵? = 1 ¥ 0 B ࠵? ࠵? ࠵? " ࠵? ࠵? ࠵?࠵? = −࠵? " ࠵? ࠵? D ¥ ࠵? = −[ ࠵? ࠵? ¥ ¥ ࠵? ࠵? ࠵? ࠵? ] = −[࠵? − ࠵?] = ࠵? 2. Derive the expression for its cumulative distribution function (cdf).
࠵?࠵?࠵? = B ࠵?(࠵?)࠵?࠵? + 0 ࠵?࠵?࠵? = B ࠵? ࠵? ࠵? " ࠵? ࠵? ࠵?࠵? = −࠵? " ࠵? ࠵? D ࠵? ࠵? = −[ ࠵? ࠵? ࠵? ࠵? ࠵? ࠵? ࠵? ࠵? ࠵? ] = − H ࠵? ࠵? ࠵? ࠵? − ࠵? I ࠵?࠵?࠵? = ࠵? − ࠵? ࠵? ࠵? ࠵? = ࠵? − ࠵? " ࠵? ࠵? 3. Derive the expression for its complementary cumulative distribution function (ccdf). ࠵?࠵?࠵?࠵? = 1 − ࠵?࠵?࠵? ࠵?࠵?࠵?࠵? = 1 − ) 1 − ࠵? " + , + ࠵?࠵?࠵?࠵? = ࠵? " ࠵? ࠵? Or: ࠵?࠵?࠵?࠵? = B ࠵?(࠵?)࠵?࠵? ¥ + ࠵?࠵?࠵?࠵? = B 1 ࠵? ࠵? " + , ࠵?࠵? = −࠵? " + , D ¥ ࠵? = −[ 1 ࠵? ¥ ¥ + 1 ࠵? + , ] = − H− 1 ࠵? + , I ࠵?࠵?࠵?࠵? = ࠵? ࠵? ࠵? ࠵? = ࠵? " ࠵? ࠵? Use a plotting routine to plot the pdf, cdf, and ccdf for a value of C = 2. V PDF CDF CCDF 0 0.5 0 1 0.5 0.38940039 0.22119922 0.77880078 1 0.30326533 0.39346934 0.60653066 1.5 0.23618328 0.52763345 0.47236655 2 0.18393972 0.63212056 0.36787944 2.5 0.1432524 0.7134952 0.2865048 3 0.11156508 0.77686984 0.22313016 3.5 0.08688697 0.82622606 0.17377394 4 0.06766764 0.86466472 0.13533528 4.5 0.05269961 0.89460078 0.10539922 5 0.0410425 0.917915 0.082085 5.5 0.03196393 0.93607214 0.06392786 6 0.02489353 0.95021293 0.04978707 6.5 0.0193871 0.96122579 0.03877421 7 0.01509869 0.96980262 0.03019738 7.5 0.01175887 0.97648225 0.02351775
8 0.00915782 0.98168436 0.01831564 8.5 0.00713212 0.98573577 0.01426423 9 0.0055545 0.988891 0.011109 9.5 0.00432585 0.9913483 0.0086517 10 0.00336897 0.99326205 0.00673795 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 Probability v Graph of PDF, CDF and CCDF PDF CDF CCDF
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