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Che320 Midterm Review

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CHE320 - Statistical Modeling & Quality Enhancement
Sample Variance and Standard Deviation, Percentile rank. Discrete Random Variables
∑𝑛𝑖=1(𝑥𝑖 − x̄ )2
𝑠2 = 𝑎𝑛𝑑 𝑠 = √𝑠 2 Probability mass function:
𝑛−1 𝑛
(𝑛+1)𝑝 1) 𝑓(𝑥𝑖 ) ≥ 0 2) ∑ 𝑓(𝑥𝑖 ) = 1
Rank (location) of pth percentile:
100 𝑖=1

Addition Rule (Probability of a Union) 3) 𝑓(𝑥𝑖 ) = 𝑃(𝑋 = 𝑥𝑖 )


𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵) Cumulative Distribution Function:
Μutually Exclusive Events
𝐹(𝑥) = 𝑃(𝑋 ≤ 𝑥) = ∑ 𝑓(𝑥𝑖 )
𝑃(𝐴 ∩ B) = 0 𝑥𝑖 ≤𝑥
Mean and Variance:
Independent Events (n events)
𝜇 = 𝐸(𝑋) = ∑ 𝑥𝑓(𝑥)
𝑃(𝐴1 ∩ 𝐴2 ∩ … ∩ 𝐴𝑛 ) = 𝑃(𝐴1 ) ∙ 𝑃(𝐴2 ) ∙ … ∙ 𝑃(𝐴𝑛 )
𝑥
𝜎 2 = 𝑉(𝑋) = 𝐸(𝑋 − 𝜇)2 = ∑𝑥(𝑥 − 𝜇)2 𝑓(𝑥)

𝑜𝑟 𝑉(𝑥) = 𝐸(𝑋 2 ) − 𝜇 2 = ∑𝑥 𝑥 2 𝑓(𝑥) − 𝜇 2

Correlation between X and Y Binomial Distribution


𝐶𝑜𝑣(𝑋, 𝑌) 𝜎𝑋𝑌
Population: 𝜌𝑋𝑌 = = Probability mass function:
√𝑉(𝑋)𝑉(𝑌) 𝜎𝑋 𝜎𝑌
𝑛
𝑛 𝑓(𝑥) = ( ) 𝑝 𝑥 (1 − 𝑝)𝑛−𝑥 𝑤ℎ𝑒𝑟𝑒 𝑥 = 0,1, … , 𝑛
𝑆𝑥𝑦 𝑥
Sample: 𝑟𝑋𝑌 = , 𝑆𝑥𝑦 = ∑(𝑥𝑖 − 𝑥̅ )(𝑦𝑖 − 𝑦̅) 0≤𝑝≤1
√𝑆𝑥𝑥 𝑆𝑦𝑦 𝑖=1 Mean and Variance:
𝑛 𝑛 𝜇 = 𝐸(𝑋) = 𝑛𝑝
2
𝑆𝑥𝑥 = ∑(𝑥𝑖 − 𝑥̅ )2 , 𝑆𝑦𝑦 = ∑(𝑦𝑖 − 𝑦
̅)
𝑖=1 𝑖=1 𝜎 2 = 𝑉(𝑋) = 𝑛𝑝(1 − 𝑝)
Covariance between two random variables X and Y, based on
their joint distribution f(x,y):
𝐶𝑜𝑣(𝑋, 𝑌) = 𝐸[(𝛸 − 𝜇𝛸 )(𝛶 − 𝜇𝛶 )]= 𝐸(𝑋𝑌) − 𝜇𝑋 𝜇𝛶

If X and Y are independent: 𝐶𝑜𝑣(𝑋, 𝑌) = 0

Linear combination of two random variables:


If Y  c0  c1 X1  c2 X 2 then E(Y )  c0  c1E( X1 )  c2 E( X 2 )

V (Y )  c12V ( X 1 )  c2 2V ( X 2 )  2c1c2Cov  X 1 , X 2 
Continuous Random Variables Cumulative Distribution Function:
𝑥

Probability density function: 𝐹(𝑥) = 𝑃(𝑋 ≤ 𝑥) = ∫ 𝑓(𝑢)𝑑𝑢


−∞
1)𝑓(𝑥𝑖 ) ≥ 0 Mean, Variance and Standard Deviation:

∞ 𝜇 = 𝐸(𝑋) = ∫ 𝑥𝑓(𝑥)𝑑𝑥
2) ∫ 𝑓(𝑥)𝑑𝑥 = 1 −∞

−∞
𝜎 = 𝑉(𝑋) = ∫ (𝑥 − 𝜇)2 𝑓(𝑥) 𝑑𝑥
2
𝑏
3)𝑃(𝑎 ≤ 𝑋 ≤ 𝑏) = ∫𝑎 𝑓(𝑥)𝑑𝑥 = area under the curve −∞

= ∫ 𝑥 2 𝑓(𝑥)𝑑𝑥 − 𝜇 2
−∞

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Standard Normal Random Variable
𝑋−𝜇
𝑍= 𝑎𝑛𝑑 𝑍~𝑁(0,1), 𝑤ℎ𝑒𝑟𝑒 − ∞ < 𝑧 < ∞
𝜎
Z- Score for the Sample Mean
X̄ − 𝜇𝛸̅ X̄ − 𝜇 𝜎
𝑍= = 𝜎 𝑤ℎ𝑒𝑟𝑒 𝜇𝛸̅ = 𝜇 𝑎𝑛𝑑 𝜎𝛸̅ =
𝜎𝛸̅ √𝑛
√𝑛
Independent Random Variables (𝑋1 , 𝑋2 , … , 𝑋𝑛 )
𝑃(𝑋1 ∈ 𝐸1, 𝑋2 ∈ 𝐸2, … , 𝑋𝑛 ∈ 𝐸𝑛 )
= 𝑃(𝑋1 ∈ 𝐸1 ) ∙ 𝑃(𝑋2 ∈ 𝐸2 ) ∙ … ∙ 𝑃(𝑋𝑛 ∈ 𝐸𝑛 )

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486 APPENDIX A STATISTICAL TABLES AND CHARTS

Table I Cumulative Standard Normal Distribution

z ⫺u2


1
£(z) ⫽ P(Z ⱕ z) ⫽
2
e du
⫺⬁
12␲

Φ (z)

0 z

z ⫺0.09 ⫺0.08 ⫺0.07 ⫺0.06 ⫺0.05 ⫺0.04 ⫺0.03 ⫺0.02 ⫺0.01 ⫺0.00 z
⫺3.9 0.000033 0.000034 0.000036 0.000037 0.000039 0.000041 0.000042 0.000044 0.000046 0.000048 ⫺3.9
⫺3.8 0.000050 0.000052 0.000054 0.000057 0.000059 0.000062 0.000064 0.000067 0.000069 0.000072 ⫺3.8
⫺3.7 0.000075 0.000078 0.000082 0.000085 0.000088 0.000092 0.000096 0.000100 0.000104 0.000108 ⫺3.7
⫺3.6 0.000112 0.000117 0.000121 0.000126 0.000131 0.000136 0.000142 0.000147 0.000153 0.000159 ⫺3.6
⫺3.5 0.000165 0.000172 0.000179 0.000185 0.000193 0.000200 0.000208 0.000216 0.000224 0.000233 ⫺3.5
⫺3.4 0.000242 0.000251 0.000260 0.000270 0.000280 0.000291 0.000302 0.000313 0.000325 0.000337 ⫺3.4
⫺3.3 0.000350 0.000362 0.000376 0.000390 0.000404 0.000419 0.000434 0.000450 0.000467 0.000483 ⫺3.3
⫺3.2 0.000501 0.000519 0.000538 0.000557 0.000577 0.000598 0.000619 0.000641 0.000664 0.000687 ⫺3.2
⫺3.1 0.000711 0.000736 0.000762 0.000789 0.000816 0.000845 0.000874 0.000904 0.000935 0.000968 ⫺3.1
⫺3.0 0.001001 0.001035 0.001070 0.001107 0.001144 0.001183 0.001223 0.001264 0.001306 0.001350 ⫺3.0
⫺2.9 0.001395 0.001441 0.001489 0.001538 0.001589 0.001641 0.001695 0.001750 0.001807 0.001866 ⫺2.9
⫺2.8 0.001926 0.001988 0.002052 0.002118 0.002186 0.002256 0.002327 0.002401 0.002477 0.002555 ⫺2.8
⫺2.7 0.002635 0.002718 0.002803 0.002890 0.002980 0.003072 0.003167 0.003264 0.003364 0.003467 ⫺2.7
⫺2.6 0.003573 0.003681 0.003793 0.003907 0.004025 0.004145 0.004269 0.004396 0.004527 0.004661 ⫺2.6
⫺2.5 0.004799 0.004940 0.005085 0.005234 0.005386 0.005543 0.005703 0.005868 0.006037 0.006210 ⫺2.5
⫺2.4 0.006387 0.006569 0.006756 0.006947 0.007143 0.007344 0.007549 0.007760 0.007976 0.008198 ⫺2.4
⫺2.3 0.008424 0.008656 0.008894 0.009137 0.009387 0.009642 0.009903 0.010170 0.010444 0.010724 ⫺2.3
⫺2.2 0.011011 0.011304 0.011604 0.011911 0.012224 0.012545 0.012874 0.013209 0.013553 0.013903 ⫺2.2
⫺2.1 0.014262 0.014629 0.015003 0.015386 0.015778 0.016177 0.016586 0.017003 0.017429 0.017864 ⫺2.1
⫺2.0 0.018309 0.018763 0.019226 0.019699 0.020182 0.020675 0.021178 0.021692 0.022216 0.022750 ⫺2.0
⫺1.9 0.023295 0.023852 0.024419 0.024998 0.025588 0.026190 0.026803 0.027429 0.028067 0.028717 ⫺1.9
⫺1.8 0.029379 0.030054 0.030742 0.031443 0.032157 0.032884 0.033625 0.034379 0.035148 0.035930 ⫺1.8
⫺1.7 0.036727 0.037538 0.038364 0.039204 0.040059 0.040929 0.041815 0.042716 0.043633 0.044565 ⫺1.7
⫺1.6 0.045514 0.046479 0.047460 0.048457 0.049471 0.050503 0.051551 0.052616 0.053699 0.054799 ⫺1.6
⫺1.5 0.055917 0.057053 0.058208 0.059380 0.060571 0.061780 0.063008 0.064256 0.065522 0.066807 ⫺1.5
⫺1.4 0.068112 0.069437 0.070781 0.072145 0.073529 0.074934 0.076359 0.077804 0.079270 0.080757 ⫺1.4
⫺1.3 0.082264 0.083793 0.085343 0.086915 0.088508 0.090123 0.091759 0.093418 0.095098 0.096801 ⫺1.3
⫺1.2 0.098525 0.100273 0.102042 0.103835 0.105650 0.107488 0.109349 0.111233 0.113140 0.115070 ⫺1.2
⫺1.1 0.117023 0.119000 0.121001 0.123024 0.125072 0.127143 0.129238 0.131357 0.133500 0.135666 ⫺1.1
⫺1.0 0.137857 0.140071 0.142310 0.144572 0.146859 0.149170 0.151505 0.153864 0.156248 0.158655 ⫺1.0
⫺0.9 0.161087 0.163543 0.166023 0.168528 0.171056 0.173609 0.176185 0.178786 0.181411 0.184060 ⫺0.9
⫺0.8 0.186733 0.189430 0.192150 0.194894 0.197662 0.200454 0.203269 0.206108 0.208970 0.211855 ⫺0.8
⫺0.7 0.214764 0.217695 0.220650 0.223627 0.226627 0.229650 0.232695 0.235762 0.238852 0.241964 ⫺0.7
⫺0.6 0.245097 0.248252 0.251429 0.254627 0.257846 0.261086 0.264347 0.267629 0.270931 0.274253 ⫺0.6
⫺0.5 0.277595 0.280957 0.284339 0.287740 0.291160 0.294599 0.298056 0.301532 0.305026 0.308538 ⫺0.5
⫺0.4 0.312067 0.315614 0.319178 0.322758 0.326355 0.329969 0.333598 0.337243 0.340903 0.344578 ⫺0.4
⫺0.3 0.348268 0.351973 0.355691 0.359424 0.363169 0.366928 0.370700 0.374484 0.378281 0.382089 ⫺0.3
⫺0.2 0.385908 0.389739 0.393580 0.397432 0.401294 0.405165 0.409046 0.412936 0.416834 0.420740 ⫺0.2
⫺0.1 0.424655 0.428576 0.432505 0.436441 0.440382 0.444330 0.448283 0.452242 0.456205 0.460172 ⫺0.1
0.0 0.464144 0.468119 0.472097 0.476078 0.480061 0.484047 0.488033 0.492022 0.496011 0.500000 0.0

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APPENDIX A 487

Table I Cumulative Standard Normal Distribution (continued)

z ⫺u2


1
£(z) ⫽ P(Z ⱕ z) ⫽
2
e du
⫺⬁
12␲

Φ (z)

0 z

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 z
0.0 0.500000 0.503989 0.507978 0.511967 0.515953 0.519939 0.523922 0.527903 0.531881 0.535856 0.0
0.1 0.539828 0.543795 0.547758 0.551717 0.555760 0.559618 0.563559 0.567495 0.571424 0.575345 0.1
0.2 0.579260 0.583166 0.587064 0.590954 0.594835 0.598706 0.602568 0.606420 0.610261 0.614092 0.2
0.3 0.617911 0.621719 0.625516 0.629300 0.633072 0.636831 0.640576 0.644309 0.648027 0.651732 0.3
0.4 0.655422 0.659097 0.662757 0.666402 0.670031 0.673645 0.677242 0.680822 0.684386 0.687933 0.4
0.5 0.691462 0.694974 0.698468 0.701944 0.705401 0.708840 0.712260 0.715661 0.719043 0.722405 0.5
0.6 0.725747 0.729069 0.732371 0.735653 0.738914 0.742154 0.745373 0.748571 0.751748 0.754903 0.6
0.7 0.758036 0.761148 0.764238 0.767305 0.770350 0.773373 0.776373 0.779350 0.782305 0.785236 0.7
0.8 0.788145 0.791030 0.793892 0.796731 0.799546 0.802338 0.805106 0.807850 0.810570 0.813267 0.8
0.9 0.815940 0.818589 0.821214 0.823815 0.826391 0.828944 0.831472 0.833977 0.836457 0.838913 0.9
1.0 0.841345 0.843752 0.846136 0.848495 0.850830 0.853141 0.855428 0.857690 0.859929 0.862143 1.0
1.1 0.864334 0.866500 0.868643 0.870762 0.872857 0.874928 0.876976 0.878999 0.881000 0.882977 1.1
1.2 0.884930 0.886860 0.888767 0.890651 0.892512 0.894350 0.896165 0.897958 0.899727 0.901475 1.2
1.3 0.903199 0.904902 0.906582 0.908241 0.909877 0.911492 0.913085 0.914657 0.916207 0.917736 1.3
1.4 0.919243 0.920730 0.922196 0.923641 0.925066 0.926471 0.927855 0.929219 0.930563 0.931888 1.4
1.5 0.933193 0.934478 0.935744 0.936992 0.938220 0.939429 0.940620 0.941792 0.942947 0.944083 1.5
1.6 0.945201 0.946301 0.947384 0.948449 0.949497 0.950529 0.951543 0.952540 0.953521 0.954486 1.6
1.7 0.955435 0.956367 0.957284 0.958185 0.959071 0.959941 0.960796 0.961636 0.962462 0.963273 1.7
1.8 0.964070 0.964852 0.965621 0.966375 0.967116 0.967843 0.968557 0.969258 0.969946 0.970621 1.8
1.9 0.971283 0.971933 0.972571 0.973197 0.973810 0.974412 0.975002 0.975581 0.976148 0.976705 1.9
2.0 0.977250 0.977784 0.978308 0.978822 0.979325 0.979818 0.980301 0.980774 0.981237 0.981691 2.0
2.1 0.982136 0.982571 0.982997 0.983414 0.983823 0.984222 0.984614 0.984997 0.985371 0.985738 2.1
2.2 0.986097 0.986447 0.986791 0.987126 0.987455 0.987776 0.988089 0.988396 0.988696 0.988989 2.2
2.3 0.989276 0.989556 0.989830 0.990097 0.990358 0.990613 0.990863 0.991106 0.991344 0.991576 2.3
2.4 0.991802 0.992024 0.992240 0.992451 0.992656 0.992857 0.993053 0.993244 0.993431 0.993613 2.4
2.5 0.993790 0.993963 0.994132 0.994297 0.994457 0.994614 0.994766 0.994915 0.995060 0.995201 2.5
2.6 0.995339 0.995473 0.995604 0.995731 0.995855 0.995975 0.996093 0.996207 0.996319 0.996427 2.6
2.7 0.996533 0.996636 0.996736 0.996833 0.996928 0.997020 0.997110 0.997197 0.997282 0.997365 2.7
2.8 0.997445 0.997523 0.997599 0.997673 0.997744 0.997814 0.997882 0.997948 0.998012 0.998074 2.8
2.9 0.998134 0.998193 0.998250 0.998305 0.998359 0.998411 0.998462 0.998511 0.998559 0.998605 2.9
3.0 0.998650 0.998694 0.998736 0.998777 0.998817 0.998856 0.998893 0.998930 0.998965 0.998999 3.0
3.1 0.999032 0.999065 0.999096 0.999126 0.999155 0.999184 0.999211 0.999238 0.999264 0.999289 3.1
3.2 0.999313 0.999336 0.999359 0.999381 0.999402 0.999423 0.999443 0.999462 0.999481 0.999499 3.2
3.3 0.999517 0.999533 0.999550 0.999566 0.999581 0.999596 0.999610 0.999624 0.999638 0.999650 3.3
3.4 0.999663 0.999675 0.999687 0.999698 0.999709 0.999720 0.999730 0.999740 0.999749 0.999758 3.4
3.5 0.999767 0.999776 0.999784 0.999792 0.999800 0.999807 0.999815 0.999821 0.999828 0.999835 3.5
3.6 0.999841 0.999847 0.999853 0.999858 0.999864 0.999869 0.999874 0.999879 0.999883 0.999888 3.6
3.7 0.999892 0.999896 0.999900 0.999904 0.999908 0.999912 0.999915 0.999918 0.999922 0.999925 3.7
3.8 0.999928 0.999931 0.999933 0.999936 0.999938 0.999941 0.999943 0.999946 0.999948 0.999950 3.8
3.9 0.999952 0.999954 0.999956 0.999958 0.999959 0.999961 0.999963 0.999964 0.999966 0.999967 3.9

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