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Unformatted text preview: otal area underneath any p.d.f. is always 1 • We need to multiply the p.d.f. by nΔ, so that the total area underneath the “scaled” p.d.f. is alway nΔ 32+45",1),&6)78',/#67)9:6:;:)4;)<=94&#&>,/)?2+"@%>4&) '!" &!" “scaled” p.d.f. of an exponential !"#$%#&'() %!" $!" #!" !" !)$" !)&" !)(" !)*" #" #)$" #)&" #)(" #)*" $" $)$" $)&" $)(" $)*" %" *&+#","".,/)01#2) 11 • Bar Plot = histogram for discrete data Bar Plots • • For each discrete value, we compare the fraction of times that we observe that value vs. the probability mass function (p.m.f.) at that point. No need to worry about interval or scaling • Example (Manufacturing): For each of the 117 wafers, we record the number of defects 0.35 the wafer, and compare it with the geometric distribution on
0.3 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 fraction in data p.m.f. of geometric distribution fraction of time observed Fraction of wafers Bar plot of fraction of defects vs. Geometric p.m.f. # of defects on the wafer number of defects 12 • • • QQ (QuantileQuantile) Plot
A useful tool for comparing the data with the cumulative distribution (c.d.f.) of a hypothesize distribution QQ plots can be preferred over histograms because constructing QQ plots does not require picking bin intervals Suppose our data consist of the observations X1, X2, ...., Xn and we hypothesize that the data come from a distribution with c.d.f. F(⋅) • • • Goal: Determine how well the c.d.f F(⋅) ﬁts the data Suppose we sort the n data points in an increasing order. Let’s call these ordered observations Y1,Y2, ...,Yn • • Note: Y1 ≤ Y2 ≤⋅⋅⋅≤ Yn Fraction of the observations that are less than or equal to Yj is j/n
j − 0.5 F (Yj ) ≈ n
13 Key Idea: If Y1,..., Yn come from a distribution with c.d.f. F(⋅), then we would...
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This note was uploaded on 10/26/2010 for the course OR&IE 5580 at Cornell University (Engineering School).
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