Anonymous

### Question Description

Study Materials:

• Multiple random variables (spreadsheet posted)
• Multiple random variables – Example
• Random variables and applications
• Supporting material: MIT Open Courseware: Lectures 5 – 10, Link

Assignment:

Wave height (H) and period (Tp) are modeled using a bivariate normal distribution:

Using the multiple random variables spreadsheet posted, please perform a sensitivity analysis by changing the correlation coefficient (r) and discuss your observations (both for positive and negative correlations) regarding the shape of the joint pdf

### Unformatted Attachment Preview

MULTIPLE RANDOM VARIABLES Duration, x (hr) 6 6 6 8 8 8 10 10 10 12 12 12 Productivity, y (%) 50 70 90 50 70 90 50 70 90 50 70 90 x 6 8 10 12 Marginal PMF pX(x) 0.122 0.432 0.317 0.129 1.0 y 50 70 90 Marginal PMF p(y) 0.180 0.475 0.345 1.0 y 50 70 90 Conditional PMF pY|x(y|x=8) 0.0833 0.500 0.417 1.0 Number of Observations 2 5 10 5 30 25 8 25 11 10 6 2 139 Joint PMF pX,Y(x,y) 0.014 0.036 0.072 0.036 0.216 0.180 0.058 0.180 0.079 0.072 0.043 0.014 1.0 MULTIPLE RANDOM VARIABLES Joint PMF 0.25 0.20 0.15 pX,Y(x,y) 0.10 0.05 0.00 90 70 6 8 X (hr) 50 10 12 Y (%) MULTIPLE RANDOM VARIABLES Marginal PMF 0.5 0.4 0.3 pX(x) 0.2 0.1 0.0 6 8 X (hr) 10 12 MULTIPLE RANDOM VARIABLES Marginal PMF 0.5 0.4 0.3 pY(y) 0.2 0.1 0.0 50 70 Y (%) 90 MULTIPLE RANDOM VARIABLES Conditional PMF pY|x(y|x = 8 hr) 0.5 0.4 0.3 0.2 0.1 0.0 50 70 Y (%) 90 MULTIPLE RANDOM VARIABLES Joint Probability Density Function fX,Y (x , y) Joint Cumulative Distribution Function a F X,Y (x , y ) = P(X  a  Y  b) = b   f X,Y (x, y )dydx − − Marginal Probability Density Functions  f X (x) =  f X,Y (x, y)dy −  f Y (y ) =  f X,Y (x, y)dx − Conditional Probability Density Functions fX Y (x y) = fX,Y ( x, y) fY X (y x) = fX,Y ( x, y) fY (y ) fX (x ) or fX,Y ( x, y) = fX Y ( x y )fY (y ) or fX,Y ( x, y) = fY X ( y x )fX ( x) MULTIPLE RANDOM VARIABLES Covariance and Correlation   E(XY) =   xyf X,Y (x , y )dydx − − COV( X, Y) = E[( X −  X )(Y − Y )] = E(XY) − E( X)E(Y) = COV( X, Y) XY −1.0    1.0 Statistical Independence fX,Y (x , y) = fX ( x)fY ( y) COV( X, Y) = 0.0 and  = 0.0 MULTIPLE RANDOM VARIABLES Example Bivariate Normal Distribution 2  2 1  y −  Y − ( Y  X )( x−  X )    1  x − X   −    −  2 2 1  Y 1−   1 e 2 X   f X,Y (x , y ) =  e   2    2 Y 1 −    2 X    fX,Y (x , y) = fY X (y x) fX (x) Note: fY X (y x) is normal with E(Y X = x) =  Y + ( Y X )(x −  X ) Var (Y X = x) = 2Y (1 − 2 ) MULTIPLE RANDOM VARIABLES Multivariate Normal Distribution 0 1 0 1 -0.8 0.30 0.25 0.20 fX,Y(x,y) 0.15 0.10 0.05 0.00 3.0 0 -1 .5 0 3.00 1.50 x 0.0 0 0.00 -1.50 1.5 0 -3.00 X X Y Y X,Y -3 .0 0 y Example of Construction Crew Performance Duration, x Productivity, y Number of (hr) 6 6 6 8 8 8 10 10 10 12 12 12 (%) 50 70 90 50 70 90 50 70 90 50 70 90 Observations 2 5 10 5 30 25 8 25 11 10 6 2 139 x (hr) 6 8 10 12 Marginal PMF pX(x) 0.122 0.432 0.317 0.129 1.0 y (%) 50 70 90 Marginal PMF p(y) 0.180 0.475 0.345 1.0 y (%) 50 70 90 Conditional PMF pY|x(y|x=8 hr) 0.0833 0.500 0.417 1.0 Joint PMF pX,Y(x,y) 0.014 0.036 0.072 0.036 0.216 0.180 0.058 0.180 0.079 0.072 0.043 0.014 1.0 m X (hr) m Y (%) s X2 (hr2) s X (hr) s Y2 (% 2) s Y (%) COV(X,Y) (hr-%) rX,Y E(X) (hr) 0.086330935 0.215827338 0.431654676 0.287769784 1.726618705 1.438848921 0.575539568 1.798561151 0.791366906 0.863309353 0.517985612 0.172661871 8.91 E(Y) (%) 0.71942446 2.51798561 6.47482014 1.79856115 15.1079137 16.1870504 2.87769784 12.5899281 7.12230216 3.5971223 3.02158273 1.29496403 73.3 2 E[(X-mX) ] (hr2) 0.121548142 0.303870355 0.607740711 0.029557432 0.17734459 0.147787158 0.068822867 0.21507146 0.094631443 0.688481873 0.413089124 0.137696375 3.01 2 E[(Y-mY) ] (%2) 7.817639062 0.393950147 20.04156211 19.54409765 2.363700882 50.10390528 31.27055625 1.969750735 22.04571832 39.08819531 0.472740176 4.008312423 199 Joint PMF E[(X-mX)(Y-mY)] (hr-%) 0.974792031 0.345990999 -3.489996161 0.760048242 0.647448503 -2.721160373 -1.46701375 -0.650874156 1.444374649 -5.187630859 -0.441909295 0.742919975 -9.04 0.25 0.20 pX,Y(x,y) 0.15 0.10 0.05 0.00 90 8.91 73.3 3.01 1.73 199 14.1 -9.04 -0.37 6 70 8 50 10 x (hr) y (%) 12 Marginal PMF pX(x) Marginal PMF 0.5 0.5 0.4 0.4 0.3 pY(y) 0.3 0.2 0.2 0.1 0.1 0.0 0.0 6 8 50 10 70 12 x (hr) y (%) Conditional PMF 0.5 0.4 0.3 pY|x(y|x = 8 hr) 0.2 0.1 0.0 50 70 y (%) 90 90 Multivariate Normal Distribution 7.8E-14 4.5E-13 2.4E-12 1.2E-11 5.8E-11 2.6E-10 1.1E-09 4.2E-09 1.6E-08 5.4E-08 1.8E-07 5.4E-07 1.6E-06 4.2E-06 1.1E-05 2.5E-05 5.7E-05 0.00012 0.00024 0.00044 0.00076 0.00125 0.00193 0.00279 0.00378 0.00483 0.00579 0.00652 0.00689 0.00685 0.0064 0.00561 0.00462 0.00358 0.0026 4.7E-13 2.6E-12 1.3E-11 6.3E-11 2.8E-10 1.2E-09 4.8E-09 1.8E-08 6.3E-08 2.1E-07 6.4E-07 1.9E-06 5.1E-06 1.3E-05 3.2E-05 7.2E-05 0.00015 0.00031 0.00058 0.00102 0.00169 0.00263 0.00386 0.0053 0.00685 0.00832 0.00948 0.01016 0.01022 0.00966 0.00858 0.00716 0.00561 0.00413 0.00286 -2.40 3.7E-17 2.8E-16 2E-15 1.3E-14 8.2E-14 4.8E-13 2.7E-12 1.4E-11 6.7E-11 3.1E-10 x 1.3E-09 5.3E-09 2E-08 7.1E-08 2.4E-07 7.5E-07 2.2E-06 6.1E-06 1.6E-05 3.9E-05 8.9E-05 0.00019 0.00039 0.00074 0.00132 0.00222 0.00351 0.0052 0.00725 0.00948 0.01165 0.01346 0.0146 0.01487 0.01424 0.0128 0.01081 0.00858 0.0064 0.00448 0.00295 -2.25 2.7E-16 1.9E-15 1.3E-14 8.1E-14 4.8E-13 2.7E-12 1.4E-11 7E-11 3.2E-10 1.4E-09 5.7E-09 2.2E-08 7.9E-08 2.7E-07 8.5E-07 2.5E-06 7.1E-06 1.9E-05 4.6E-05 0.00011 0.00023 0.00048 0.00093 0.00168 0.00286 0.00456 0.00685 0.00966 0.0128 0.01593 0.01862 0.02045 0.0211 0.02045 0.01862 0.01593 0.0128 0.00966 0.00685 0.00456 0.00286 0.00 0.30 0.25 -2.70 -2.55 0.20 fX,Y6E-19 (x,y) 4.9E-18 0.15 5E-18 3.9E-17 3.9E-17 2.9E-16 0.10 2.9E-16 2E-15 0.05 2E-15 1.3E-14 0.00 1.3E-14 8.1E-14 1.50 -2.85 7E-20 6.1E-19 5E-18 3.9E-17 2.8E-16 1.9E-15 1.2E-14 7.4E-14 4.2E-13 2.2E-12 1.1E-11 5.1E-11 2.3E-10 9.4E-10 3.6E-09 1.3E-08 4.5E-08 1.5E-07 4.4E-07 1.3E-06 3.3E-06 8.4E-06 2E-05 4.4E-05 9.1E-05 0.00018 0.00032 0.00056 0.0009 0.00137 0.00196 0.00263 0.00332 0.00393 0.00437 0.00456 0.00448 0.00413 0.00358 0.00291 0.00222 -2.10 1.8E-15 1.2E-14 7.8E-14 4.7E-13 2.7E-12 1.4E-11 7E-11 3.3E-10 1.4E-09 6E-09 2.3E-08 8.5E-08 2.9E-07 9.3E-07 2.8E-06 8E-06 2.1E-05 5.4E-05 0.00013 0.00028 0.00058 0.00113 0.00208 0.00358 0.00579 0.0088 0.01256 0.01685 0.02124 0.02514 0.02796 0.02921 0.02867 0.02643 0.02289 0.01862 0.01424 0.01022 0.00689 0.00437 0.0026 3.00 -3.00 7.6E-21 7E-20 6E-19 4.9E-18 3.7E-17 2.7E-16 1.8E-15 1.1E-14 6.7E-14 3.8E-13 2E-12 9.7E-12 4.5E-11 1.9E-10 8E-10 3E-09 1.1E-08 3.7E-08 1.2E-07 3.5E-07 9.9E-07 2.6E-06 6.4E-06 1.5E-05 3.3E-05 6.7E-05 0.00013 0.00023 0.0004 0.00064 0.00096 0.00135 0.00179 0.00222 0.0026 0.00286 0.00295 0.00286 0.0026 0.00222 0.00179 -3.00 x,y -3.00 -2.85 -2.70 -2.55 -2.40 -2.25 -2.10 -1.95 -1.80 -1.65 -1.50 -1.35 -1.20 -1.05 -0.90 -0.75 -0.60 -0.45 -0.30 -0.15 0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20 1.35 1.50 1.65 1.80 1.95 2.10 2.25 2.40 2.55 2.70 2.85 3.00 0 1 0 1 -0.8 -1.50 mX sX mY sY r X,Y -1.95 1.1E-14 7.4E-14 4.5E-13 2.6E-12 1.4E-11 7E-11 3.3E-10 1.5E-09 6.1E-09 2.4E-08 8.9E-08 3.1E-07 1E-06 3.1E-06 8.9E-06 2.4E-05 6.1E-05 0.00014 0.00032 0.00068 0.00135 0.0025 0.00437 0.00716 0.01102 0.01593 0.02164 0.02761 0.0331 0.03727 0.03943 0.03918 0.03658 0.03208 0.02643 0.02045 0.01487 0.01016 0.00652 0.00393 0.00222 -1.80 6.7E-14 4.2E-13 2.4E-12 1.3E-11 6.7E-11 3.2E-10 1.4E-09 6.1E-09 y 2.4E-08 9.1E-08 3.2E-07 1.1E-06 3.3E-06 9.6E-06 2.6E-05 6.7E-05 0.00016 0.00037 0.00078 0.00157 0.00295 0.0052 0.00863 0.01346 0.0197 0.0271 0.03501 0.0425 0.04846 0.05191 0.05223 0.04938 0.04385 0.03658 0.02867 0.0211 0.0146 0.00948 0.00579 0.00332 0.00179 -1.65 3.8E-13 2.2E-12 1.2E-11 6.3E-11 3.1E-10 1.4E-09 6E-09 2.4E-08 9.1E-08 3.2E-07 1.1E-06 3.4E-06 1E-05 2.8E-05 7.2E-05 0.00018 0.00041 0.00088 0.00178 0.00338 0.00605 0.01016 0.01603 0.02376 0.0331 0.0433 0.05322 0.06145 0.06665 0.06791 0.065 0.05845 0.04938 0.03918 0.02921 0.02045 0.01346 0.00832 0.00483 0.00263 0.00135 -1.50 2E-12 1.1E-11 5.8E-11 2.8E-10 1.3E-09 5.7E-09 2.3E-08 8.9E-08 3.2E-07 1.1E-06 3.5E-06 1E-05 2.9E-05 7.6E-05 0.00019 0.00044 0.00096 0.00196 0.00378 0.00685 0.01165 0.01862 0.02796 0.03943 0.05223 0.065 0.076 0.08347 0.08612 0.08347 0.076 0.065 0.05223 0.03943 0.02796 0.01862 0.01165 0.00685 0.00378 0.00196 0.00096 -1.35 9.7E-12 5.1E-11 2.6E-10 1.2E-09 5.3E-09 2.2E-08 8.5E-08 3.1E-07 1.1E-06 3.4E-06 1E-05 2.9E-05 7.8E-05 0.0002 0.00046 0.00102 0.00212 0.00413 0.00757 0.01304 0.0211 0.03208 0.04581 0.06145 0.07744 0.09167 0.10195 0.10651 0.10453 0.09637 0.08347 0.06791 0.05191 0.03727 0.02514 0.01593 0.00948 0.0053 0.00279 0.00137 0.00064 -1.20 4.5E-11 2.3E-10 1.1E-09 4.8E-09 2E-08 7.9E-08 2.9E-07 1E-06 3.3E-06 1E-05 2.9E-05 7.8E-05 0.0002 0.00047 0.00106 0.00222 0.0044 0.00816 0.01424 0.02332 0.0359 0.05191 0.07051 0.08997 0.10785 0.12144 0.12847 0.12767 0.11919 0.10453 0.08612 0.06665 0.04846 0.0331 0.02124 0.0128 0.00725 0.00386 0.00193 0.0009 0.0004 -1.05 1.9E-10 9.4E-10 4.2E-09 1.8E-08 7.1E-08 2.7E-07 9.3E-07 3.1E-06 9.6E-06 2.8E-05 7.6E-05 0.0002 0.00047 0.00107 0.00228 0.00456 0.00858 0.01515 0.02514 0.03918 0.05737 0.0789 0.10195 0.12374 0.1411 0.15114 0.15209 0.14377 0.12767 0.10651 0.08347 0.06145 0.0425 0.02761 0.01685 0.00966 0.0052 0.00263 0.00125 0.00056 0.00023 -0.90 8E-10 3.6E-09 1.6E-08 6.3E-08 2.4E-07 8.5E-07 2.8E-06 8.9E-06 2.6E-05 7.2E-05 0.00019 0.00046 0.00106 0.00228 0.00462 0.0088 0.01573 0.02643 0.04171 0.06183 0.08612 0.11267 0.13848 0.15988 0.17342 0.1767 0.16914 0.15209 0.12847 0.10195 0.076 0.05322 0.03501 0.02164 0.01256 0.00685 0.00351 0.00169 0.00076 0.00032 0.00013 -0.75 3E-09 1.3E-08 5.4E-08 2.1E-07 7.5E-07 2.5E-06 8E-06 2.4E-05 6.7E-05 0.00018 0.00044 0.00102 0.00222 0.00456 0.0088 0.01593 0.0271 0.0433 0.065 0.09167 0.12144 0.15114 0.1767 0.19407 0.20023 0.19407 0.1767 0.15114 0.12144 0.09167 0.065 0.0433 0.0271 0.01593 0.0088 0.00456 0.00222 0.00102 0.00044 0.00018 6.7E-05 -0.60 1.1E-08 4.5E-08 1.8E-07 6.4E-07 2.2E-06 7.1E-06 2.1E-05 6.1E-05 0.00016 0.00041 0.00096 0.00212 0.0044 0.00858 0.01573 0.0271 0.04385 0.06665 0.09517 0.12767 0.16089 0.19046 0.21181 0.22129 0.21718 0.20023 0.17342 0.1411 0.10785 0.07744 0.05223 0.0331 0.0197 0.01102 0.00579 0.00286 0.00132 0.00058 0.00024 9.1E-05 3.3E-05 -0.45 3.7E-08 1.5E-07 5.4E-07 1.9E-06 6.1E-06 1.9E-05 5.4E-05 0.00014 0.00037 0.00088 0.00196 0.00413 0.00816 0.01515 0.02643 0.0433 0.06665 0.09637 0.1309 0.16704 0.20023 0.22547 0.23852 0.23703 0.22129 0.19407 0.15988 0.12374 0.08997 0.06145 0.03943 0.02376 0.01346 0.00716 0.00358 0.00168 0.00074 0.00031 0.00012 4.4E-05 1.5E-05 -0.30 1.2E-07 4.4E-07 1.6E-06 5.1E-06 1.6E-05 4.6E-05 0.00013 0.00032 0.00078 0.00178 0.00378 0.00757 0.01424 0.02514 0.04171 0.065 0.09517 0.1309 0.16914 0.2053 0.23409 0.25075 0.25232 0.23852 0.21181 0.1767 0.13848 0.10195 0.07051 0.04581 0.02796 0.01603 0.00863 0.00437 0.00208 0.00093 0.00039 0.00015 5.7E-05 2E-05 6.4E-06 -0.15 3.5E-07 1.3E-06 4.2E-06 1.3E-05 3.9E-05 0.00011 0.00028 0.00068 0.00157 0.00338 0.00685 0.01304 0.02332 0.03918 0.06183 0.09167 0.12767 0.16704 0.2053 0.23703 0.2571 0.26196 0.25075 0.22547 0.19046 0.15114 0.11267 0.0789 0.05191 0.03208 0.01862 0.01016 0.0052 0.0025 0.00113 0.00048 0.00019 7.2E-05 2.5E-05 8.4E-06 2.6E-06 0.00 9.9E-07 3.3E-06 1.1E-05 3.2E-05 8.9E-05 0.00023 0.00058 0.00135 0.00295 0.00605 0.01165 0.0211 0.0359 0.05737 0.08612 0.12144 0.16089 0.20023 0.23409 0.2571 0.26526 0.2571 0.23409 0.20023 0.16089 0.12144 0.08612 0.05737 0.0359 0.0211 0.01165 0.00605 0.00295 0.00135 0.00058 0.00023 8.9E-05 3.2E-05 1.1E-05 3.3E-06 9.9E-07 0.15 2.6E-06 8.4E-06 2.5E-05 7.2E-05 0.00019 0.00048 0.00113 0.0025 0.0052 0.01016 0.01862 0.03208 0.05191 0.0789 0.11267 0.15114 0.19046 0.22547 0.25075 0.26196 0.2571 0.23703 0.2053 0.16704 0.12767 0.09167 0.06183 0.03918 0.02332 0.01304 0.00685 0.00338 0.00157 0.00068 0.00028 0.00011 3.9E-05 1.3E-05 4.2E-06 1.3E-06 3.5E-07 0.30 6.4E-06 2E-05 5.7E-05 0.00015 0.00039 0.00093 0.00208 0.00437 0.00863 0.01603 0.02796 0.04581 0.07051 0.10195 0.13848 0.1767 0.21181 0.23852 0.25232 0.25075 0.23409 0.2053 0.16914 0.1309 0.09517 0.065 0.04171 0.02514 0.01424 0.00757 0.00378 0.00178 0.00078 0.00032 0.00013 4.6E-05 1.6E-05 5.1E-06 1.6E-06 4.4E-07 1.2E-07 0.45 1.5E-05 4.4E-05 0.00012 0.00031 0.00074 0.00168 0.00358 0.00716 0.01346 0.02376 0.03943 0.06145 0.08997 0.12374 0.15988 0.19407 0.22129 0.23703 0.23852 0.22547 0.20023 0.16704 0.1309 0.09637 0.06665 0.0433 0.02643 0.01515 0.00816 0.00413 0.00196 0.00088 0.00037 0.00014 5.4E-05 1.9E-05 6.1E-06 1.9E-06 5.4E-07 1.5E-07 3.7E-08 0.60 3.3E-05 9.1E-05 0.00024 0.00058 0.00132 0.00286 0.00579 0.01102 0.0197 0.0331 0.05223 0.07744 0.10785 0.1411 0.17342 0.20023 0.21718 0.22129 0.21181 0.19046 0.16089 0.12767 0.09517 0.06665 0.04385 0.0271 0.01573 0.00858 0.0044 0.00212 0.00096 0.00041 0.00016 6.1E-05 2.1E-05 7.1E-06 2.2E-06 6.4E-07 1.8E-07 4.5E-08 1.1E-08 0.75 6.7E-05 0.00018 0.00044 0.00102 0.00222 0.00456 0.0088 0.01593 0.0271 0.0433 0.065 0.09167 0.12144 0.15114 0.1767 0.19407 0.20023 0.19407 0.1767 0.15114 0.12144 0.09167 0.065 0.0433 0.0271 0.01593 0.0088 0.00456 0.00222 0.00102 0.00044 0.00018 6.7E-05 2.4E-05 8E-06 2.5E-06 7.5E-07 2.1E-07 5.4E-08 1.3E-08 3E-09 0.90 0.00013 0.00032 0.00076 0.00169 0.00351 0.00685 0.01256 0.02164 0.03501 0.05322 0.076 0.10195 0.12847 0.15209 0.16914 0.1767 0.17342 0.15988 0.13848 0.11267 0.08612 0.06183 0.04171 0.02643 0.01573 0.0088 0.00462 0.00228 0.00106 0.00046 0.00019 7.2E-05 2.6E-05 8.9E-06 2.8E-06 8.5E-07 2.4E-07 6.3E-08 1.6E-08 3.6E-09 8E-10 1.05 0.00023 0.00056 0.00125 0.00263 0.0052 0.00966 0.01685 0.02761 0.0425 0.06145 0.08347 0.10651 0.12767 0.14377 0.15209 0.15114 0.1411 0.12374 0.10195 0.0789 0.05737 0.03918 0.02514 0.01515 0.00858 0.00456 0.00228 0.00107 0.00047 0.0002 7.6E-05 2.8E-05 9.6E-06 3.1E-06 9.3E-07 2.7E-07 7.1E-08 1.8E-08 4.2E-09 9.4E-10 1.9E-10 1.20 0.0004 0.0009 0.00193 0.00386 0.00725 0.0128 0.02124 0.0331 0.04846 0.06665 0.08612 0.10453 0.11919 0.12767 0.12847 0.12144 0.10785 0.08997 0.07051 0.05191 0.0359 0.02332 0.01424 0.00816 0.0044 0.00222 0.00106 0.00047 0.0002 7.8E-05 2.9E-05 1E-05 3.3E-06 1E-06 2.9E-07 7.9E-08 2E-08 4.8E-09 1.1E-09 2.3E-10 4.5E-11 1.35 0.00064 0.00137 0.00279 0.0053 0.00948 0.01593 0.02514 0.03727 0.05191 0.06791 0.08347 0.09637 0.10453 0.10651 0.10195 0.09167 0.07744 0.06145 0.04581 0.03208 0.0211 0.01304 0.00757 0.00413 0.00212 0.00102 0.00046 0.0002 7.8E-05 2.9E-05 1E-05 3.4E-06 1.1E-06 3.1E-07 8.5E-08 2.2E-08 5.3E-09 1.2E-09 2.6E-10 5.1E-11 9.7E-12 1.50 0.00096 0.00196 0.00378 0.00685 0.01165 0.01862 0.02796 0.03943 0.05223 0.065 0.076 0.08347 0.08612 0.08347 0.076 0.065 0.05223 0.03943 0.02796 0.01862 0.01165 0.00685 0.00378 0.00196 0.00096 0.00044 0.00019 7.6E-05 2.9E-05 1E-05 3.5E-06 1.1E-06 3.2E-07 8.9E-08 2.3E-08 5.7E-09 1.3E-09 2.8E-10 5.8E-11 1.1E-11 2E-12 1.65 0.00135 0.00263 0.00483 0.00832 0.01346 0.02045 0.02921 0.03918 0.04938 0.05845 0.065 0.06791 0.06665 0.06145 0.05322 0.0433 0.0331 0.02376 0.01603 0.01016 0.00605 0.00338 0.00178 0.00088 0.00041 0.00018 7.2E-05 2.8E-05 1E-05 3.4E-06 1.1E-06 3.2E-07 9.1E-08 2.4E-08 6E-09 1.4E-09 3.1E-10 6.3E-11 1.2E-11 2.2E-12 3.8E-13 1.80 0.00179 0.00332 0.00579 0.00948 0.0146 0.0211 0.02867 0.03658 0.04385 0.04938 0.05223 0.05191 0.04846 0.0425 0.03501 0.0271 0.0197 0.01346 0.00863 0.0052 0.00295 0.00157 0.00078 0.00037 0.00016 6.7E-05 2.6E-05 9.6E-06 3.3E-06 1.1E-06 3.2E-07 9.1E-08 2.4E-08 6.1E-09 1.4E-09 3.2E-10 6.7E-11 1.3E-11 2.4E-12 4.2E-13 6.7E-14 1.95 0.00222 0.00393 0.00652 0.01016 0.01487 0.02045 0.02643 0.03208 0.03658 0.03918 0.03943 0.03727 0.0331 0.02761 0.02164 0.01593 0.01102 0.00716 0.00437 0.0025 0.00135 0.00068 0.00032 0.00014 6.1E-05 2.4E-05 8.9E-06 3.1E-06 1E-06 3.1E-07 8.9E-08 2.4E-08 6.1E-09 1.5E-09 3.3E-10 7E-11 1.4E-11 2.6E-12 4.5E-13 7.4E-14 1.1E-14 2.10 0.0026 0.00437 0.00689 0.01022 0.01424 0.01862 0.02289 0.02643 0.02867 0.02921 0.02796 0.02514 0.02124 0.01685 0.01256 0.0088 0.00579 0.00358 0.00208 0.00113 0.00058 0.00028 0.00013 5.4E-05 2.1E-05 8E-06 2.8E-06 9.3E-07 2.9E-07 8.5E-08 2.3E-08 6E-09 1.4E-09 3.3E-10 7E-11 1.4E-11 2.7E-12 4.7E-13 7.8E-14 1.2E-14 1.8E-15 2.25 0.00286 0.00456 0.00685 0.00966 0.0128 0.01593 0.01862 0.02045 0.0211 0.02045 0.01862 0.01593 0.0128 0.00966 0.00685 0.00456 0.00286 0.00168 0.00093 0.00048 0.00023 0.00011 4.6E-05 1.9E-05 7.1E-06 2.5E-06 8.5E-07 2.7E-07 7.9E-08 2.2E-08 5.7E-09 1.4E-09 3.2E-10 7E-11 1.4E-11 2.7E-12 4.8E-13 8.1E-14 1.3E-14 1.9E-15 2.7E-16 2.40 0.00295 0.00448 0.0064 0.00858 0.01081 0.0128 0.01424 0.01487 0.0146 0.01346 0.01165 0.00948 0.00725 0.0052 0.00351 0.00222 0.00132 0.00074 0.00039 0.00019 8.9E-05 3.9E-05 1.6E-05 6.1E-06 2.2E-06 7.5E-07 2.4E-07 7.1E-08 2E-08 5.3E-09 1.3E-09 3.1E-10 6.7E-11 1.4E-11 2.7E-12 4.8E-13 8.2E-14 1.3E-14 2E-15 2.8E-16 3.7E-17 2.55 0.00286 0.00413 0.00561 0.00716 0.00858 0.00966 0.01022 0.01016 0.00948 0.00832 0.00685 0.0053 0.00386 0.00263 0.00169 0.00102 0.00058 0.00031 0.00015 7.2E-05 3.2E-05 1.3E-05 5.1E-06 1.9E-06 6.4E-07 2.1E-07 6.3E-08 1.8E-08 4.8E-09 1.2E-09 2.8E-10 6.3E-11 1.3E-11 2.6E-12 4.7E-13 8.1E-14 1.3E-14 2E-15 2.9E-16 3.9E-17 4.9E-18 2.70 0.0026 0.00358 0.00462 0.00561 0.0064 0.00685 0.00689 0.00652 0.00579 0.00483 0.00378 0.00279 0.00193 0.00125 0.00076 0.00044 0.00024 0.00012 5.7E-05 2.5E-05 1.1E-05 4.2E-06 1.6E-06 5.4E-07 1.8E-07 5.4E-08 1.6E-08 4.2E-09 1.1E-09 2.6E-10 5.8E-11 1.2E-11 2.4E-12 4.5E-13 7.8E-14 1.3E-14 2E-15 2.9E-16 3.9E-17 5E-18 6E-19 2.85 0.00222 0.00291 0.00358 0.00413 0.00448 0.00456 0.00437 0.00393 0.00332 0.00263 0.00196 0.00137 0.0009 0.00056 0.00032 0.00018 9.1E-05 4.4E-05 2E-05 8.4E-06 3.3E-06 1.3E-06 4.4E-07 1.5E-07 4.5E-08 1.3E-08 3.6E-09 9.4E-10 2.3E-10 5.1E-11 1.1E-11 2.2E-12 4.2E-13 7.4E-14 1.2E-14 1.9E-15 2.8E-16 3.9E-17 5E-18 6.1E-19 7E-20 3.00 0.00179 0.00222 0.0026 0.00286 0.00295 0.00286 0.0026 0.00222 0.00179 0.00135 0.00096 0.00064 0.0004 0.00023 0.00013 6.7E-05 3.3E-05 1.5E-05 6.4E-06 2.6E-06 9.9E-07 3.5E-07 1.2E-07 3.7E-08 1.1E-08 3E-09 8E-10 1.9E-10 4.5E-11 9.7E-12 2E-12 3.8E-13 6.7E-14 1.1E-14 1.8E-15 2.7E-16 3.7E-17 4.9E-18 6E-19 7E-20 7.6E-21 Random Variables and Applications OPRE 6301 Random Variables. . . As noted earlier, variability is omnipresent in the business world. To model variability probabilistically, we need the concept of a random variable. A random variable is a numerically valued variable which takes on different values with given probabilities. Examples: The return on an investment in a one-year period The price of an equity The number of customers entering a store The sales volume of a store on a particular day The turnover rate at your organization next year 1 Types of Random Variables. . . Discrete Random Variable: — one that takes on a countable number of possible values, e.g., • total of roll of two dice: 2, 3, . . . , 12 • number of desktops sold: 0, 1, . . . • customer count: 0, 1, . . . Continuous Random Variable: — one that takes on an uncountable number of possible values, e.g., • interest rate: 3.25%, 6.125%, . . . • task completion time: a nonnegative value • price of a stock: a nonnegative value Basic Concept: Integer or rational numbers are discrete, while real numbers are continuous. 2 Probability Distributions. . . “Randomness” of a random variable is described by a probability distribution. Informally, the probability distribution specifies the probability or likelihood for a random variable to assume a particular value. Formally, let X be a random variable and let x be a possible value of X. Then, we have two cases. Discrete: the probability mass function of X specifies P (x) ≡ P (X = x) for all possible values of x. Continuous: the probability density function of X is a function f (x) that is such that f (x) · h ≈ P (x < X ≤ x + h) for small positive h. Basic Concept: The probability mass function specifies the actual probability, while the probability density function specifies the probability rate; both can be viewed as a measure of “likelihood.” The continuous case will be discussed in Chapter 8. 3 Discrete Distributions. . . ...
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