# solutions manual random phenomena fundamentals and engineering applications of probabiltiy and...

Post on 28-Jul-2015

1.405 views

Category:

## Documents

Embed Size (px)

TRANSCRIPT

9 6 3 0 -3MedianMean4.5 4.0 3.5 3.0 2.5 2.01st Quartile 1.0978Median 2.89163rd Quartile 5.2501Maximum 9.11112.1032 3.99031.8908 4.29912.7733 4.1371A -Squared 0.27P-Value 0.653Mean 3.0467StDev 3.3200V ariance 11.0221Skewness -0.188360Kurtosis -0.456418N 50Minimum -5.1712A nderson-Darling NormalityTest95% Confidence Interv al for Mean95% Confidence Interv al for Median95% C onfidence Interv al for StDev95% Confidence IntervalsSummary for d

Figure1.1: Histogramford = YAYBdatawithsuperimposedtheoreticaldistribution12 CHAPTER1.From the data, the arithmetic average,d, is obtained asd = 3.05 (1.1)Andnow, thatthisaverageispositive, notzero, suggeststhepossibilitythatYA may be greater than YB. However conclusive evidence requires a measure ofintrinsic variability.1.3 Directly from the data in Table 1.1 in the text, we obtain yA = 75.52; yB =72.47; ands2A = 2.05;s2B = 7.64. Also directly from the table of dierences,di,generated for Exercise 1.2, we obtain:d = 3.05; howevers2d = 11.02, not 9.71.Thus, even though for the means,d = yA yBfor the variances,s2d = s2A + s2BThereasonforthisdiscrepancyisthatforthevarianceequalitytohold, YAmust be completelyindependent ofYBso that the covariance betweenYAandYBis precisely zero. While this may be true of the actual random variable, itis not always strictly the case with data. The more general expression which isvalid in all cases is as follows:s2d = s2A + s2B2sAB(1.2)wheresABisthecovariancebetweenyAandyB(seeChapters4and12). Inthis particular case, the covariance between the yA and yBdata is computed assAB = 0.67Observe that thevalue computedfors2d(11.02) isobtained by adding 2sABtos2A + s2B, as in Eq (1.2).Section1.21.4 From the data in Table 1.2 in the text,s2x = 1.2.1.5 In this case, with x = 1.02, and variance, s2x = 1.2, even though the num-bers are not exactly equal, within limits of random variation, they appear to beclose enough, suggesting the possibility that X may in fact be a Poisson randomvariable.Section1.31.6Thehistogramsobtainedwithbinsizesof0.75, shownbelow, contain10bins forYAversus 8 bins for the histogram of Fig 1.1 in the text, and 14 binsforYBversus 11 bins in Fig 1.2 in the text. These new histograms show a bitmore detail but the general features displayed for the data sets are essentiallyunchanged. When the bin sizes are expanded to 2.0, things are slightly dierent,379.5 78.0 76.5 75.0 73.5 72.0181614121086420YAFrequencyHistogramof YA (Bin size 0.75)

78.0 76.5 75.0 73.5 72.0 70.5 69.0 67.56543210YBFrequencyHistogramof YB (Bin size 0.75)

Figure1.2: HistogramforYA, YBdatawithsmallbinsize(0.75)

80 78 76 74 722520151050YAFrequencyHistogramof YA (Bin size 2.0)

79 77 75 73 71 69 6714121086420YBFrequencyHistogramof YB(Bin Size 2.0)

Figure1.3: HistogramforYA, YBdatawithlargerbinsize(2.0)4 CHAPTER1.as shown below. These histograms now contain fewer bins (5 forYAand 7 forYB); and, hence in general, show less of the true character of the data sets.1.7 The values computed from the data for yAandsAimply that the intervalof interest, yA 1.96sA, is 75.52 2.81, or (72.71, 78.33). From the frequencydistributionofTable1.3inthetext,48ofthe50pointslieinthisrange,theexcludedpointsbeing(i)thesinglepointinthe71.5172.50binand(ii)thesingle point in the 78.5179.50 bin. Thus, this interval contains 96% of the data.1.8FortheYBdata, theintervalofinterest, yB 1.96sB, is72.47 5.41, or(67.06, 77.88). From Table 1.4 in the text,we see that approximately 48 of the50 points lie in this range (excluding the 2 points in the 77.5178.50 bin). Thus,this interval also contains approximately 96% of the data.1.9 From Table 1.4 in the text, we observe that the relative frequency associatedwithx = 4 is 0.033; that associated withx = 5 is 0.017 and 0 thereafter. Theimplication is that the relative frequency associated with x > 3 = 0.050. Hence,the value ofx such that only 5% of the data exceeds this value isx = 3.1.10Using=75.52and=1.43, thetheoretical valuescomputedforthefunctioninEq1.3inthetext, (fory=72, 73, . . . , 79)areshowninthetablebelow along with the the corresponding relative frequency values from Table 1.3in the text.Theoretical RelativeYAGroup y f(y) Frequency71.51-72.50 72 0.014 0.0272.51-73.50 73 0.059 0.0473.51-74.50 74 0.159 0.1874.51-75.50 75 0.261 0.3475.51-76.50 76 0.264 0.1476.51-77.50 77 0.163 0.1677.51-78.50 78 0.062 0.1078.51-79.50 79 0.014 0.02TOTAL 50 0.996 1.00The agreement between the theoretical values and the relative frequency is rea-sonable but not perfect.1.11 This time time with = 72.47 and = 2.76 and for y = 67, 68, 69, . . . , 79,weobtainthetableshownbelowfortheYBdata(alongwiththethecorre-sponding relative frequency values from Table 1.4 in the text).5Theoretical RelativeYBGroup y f(y) Frequency66.51-67.50 67 0.020 0.0267.51-68.50 68 0.039 0.0668.51-69.50 69 0.066 0.0869.51-70.50 70 0.097 0.1670.51-71.50 71 0.125 0.0471.51-72.50 72 0.142 0.1472.51-73.50 73 0.142 0.0873.51-74.50 74 0.124 0.1274.51-75.50 75 0.095 0.1075.51-76.50 76 0.064 0.1276.51-77.50 77 0.038 0.0077.51-78.50 78 0.019 0.0478.51-79.50 79 0.009 0.00TOTAL 50 0.980 1.00Thereisreasonableagreementbetweenthetheoreticalvaluesandtherelativefrequency.1.12Using=1.02, thetheoretical valuesof thefunctionf(x|)of Eq1.4inthetextat x=0, 1, 2, . . . 6areshowninthetablebelowalongwiththecorresponding relative frequency values from Table 1.5 in the text.Theoretical RelativeX f(x| = 1.02) Frequency0 0.3606 0.3671 0.3678 0.3832 0.1876 0.1833 0.0638 0.0174 0.0163 0.0335 0.0033 0.0176 0.0006 0.000TOTAL 1.0000 1.000The agreement between the theoreticalf(x) and the data relative frequency isreasonable. (This pdf was plotted in Fig 1.6 of the text.)ApplicationProblems1.13 (i) The following is one way to generate a frequency distribution for thisdata:6 CHAPTER1.RelativeX Frequency Frequency1.00-3.00 4 0.0473.01-5.00 9 0.1065.01-7.00 11 0.1297.01-9.00 20 0.2359.01-11.00 10 0.11811.01-13.00 9 0.10613.01-15.00 3 0.03515.01-17.00 6 0.07017.01-19.00 6 0.07019.01-21.00 5 0.05921.01-23.00 1 0.01223.01-25.00 1 0.012TOTAL 85 0.999The histogram resulting from this frequency distribution is shown below wherewe observe that it is skewed to the right. Superimposed on the histogram is atheoretical gamma distribution, which ts the data quite well. The variable inquestion,time-to-publication, is(a)non-negative,(b)continuous,and(c)hasthepotential tobealargenumber(ifapapergoesthroughseveral revisionsbeforeitisnallyaccepted, orif thereviewersaretardyincompletingtheirreviews in the rst place). It is therefore not surprising that the histogram willbe skewed to the right as shown.

24 20 16 12 8 4 020151050xFrequencyShape 3.577Scale 2.830N 85Histogram of xGamma

Figure1.4: Histogramfortime-to-publicationdata(ii) From this frequency distribution and the histogram, we see that the mostpopulartime-to-publicationisintherangefrom7-9months(centeredat8months); from the relative frequency values, we note that 41/85 or 0.482 is the7fraction of the papers that took longer than this to publish.1.14(i)Aplotofthehistogramforthe20-sampleaverages, yi, generatedasprescribed is shown in the top panel of the gure below. We note the narrowerrange occupied by this data set as well as its more symmetric nature. (Super-imposed on this histogram is a the