Df for exponential fx 1 e x for x 0 inverse cdf

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Unformatted text preview: expect that Note: Subtracting 0.5 can be visualized as a “continuity correction”. The impact of this is negligible when the number of data points n is large! Q-Q Plot (cont.) j − 0.5 F (Yj ) ≈ n ⇔ Yj ≈ F −1 ￿ j − 0.5 n ￿ , j = 1, 2, . . . n • • • • This means that Yj is approximately the [(j - 0.5)/n]th quantile of F(⋅) Idea: Compare Yj with the true [(j - 0.5)/n]th quantile of F(⋅) In the Q-Q plot, we will plot the following collection of points: If Y1,Y2, ...,Yn do really come from a distribution with c.d.f. F(⋅), then we would expect the above collection of points to lie on a straight line that makes a 45°-angle with the x-axis! ￿￿ Yj , F −1 ￿ ￿ ((j − 0.5)/n) : j = 1, 2, . . . , n • This provides a visual check on how well the c.d.f. F (⋅) fits the data! 14 • • Interarrival Time Revisited Recall: Data on interarrival time of 174 customers over 90 minutes Our histogram suggests that an exponential distribution with parameter λ=1.93 (= 174/90) fits the data well. • Exponential Quantiles exponential quantiles • • 3 c.d.f. for exponential: F(x) = 1 - e-λx for x ≥ 0 Inverse c.d.f: F-1(u) = - ln (1 - u) / λ for 0 ≤ u < 1 Order the 174 observations in increasing order to obtain Y1,Y2, ...,Y174 . 2.5 2 1.5 1 0.5 0 0 Q-Q Plot (Yj , F −1 ((j − 0.5)/n)) ￿ ￿ ￿￿ 1 j − 0.5 = Yj , − ln 1 − λ n 0.5 Observed Quantiles observed quantiles 1 1.5 2 2.5 3 15 Things to Consider When Evaluating Q-Q Plot • The observed values will never fall exactly on the straight line • The ordered values are not independent because we ordered them. • This means that if one point lies above the line, it is likely that the next one does as well. • • The values at the extremes have a much higher variance than those in the middle. • So, greater departure from linearity can be acceptable towards the ends of the plot Based on these guidelines, the exponential distribution fits the data in our example quite well. 16 Parameter Estimat...
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