NBA Regular Season Over/Under Betting Analysis
Explore the empirical joint probability distributions of NBA regular season over/under betting lines and actual total points scored from 2006/07 to 2018/19. Analyze the bivariate association between over/under lines and total points, including means, variances, covariances, and correlations. Gain insights from univariate summaries and histograms presented in the data.
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Application of Empirical Joint Probability Distributions NBA Regular Season Over/Under Betting Lines and Actual Total Points 2006/07- 2018/19 Regular Seasons Data Source: covers.com
Data Description Population: N = 15749 National Basketball Association Games from 2006/07-2018/19 Seasons Variables observed for each game: X = Over/Under Betting Line set pre-game for Total Points Scored in Game Y = Total Points Actually Scored in Game Z = Over/Under Betting Range (Representing a Partition of Sample Space of Over/Under), Used in Demonstrating Expectations, Variances, and Covariances Note: all computations for Standard Deviations, Variances, and Covariances are adjusted to use N, not N-1 as divisor
Univariate Summaries Parameter X m m s2 s2 s s min LQ median UQ max skewness kurtosis Y z [OU.LO OU.HI) N(z) p(z) 202.817 156.235 12.499 170.500 193.500 202.000 211.000 245.000 0.334 -2.973 203.213 458.054 21.402 134.000 189.000 202.000 217.000 329.000 0.335 -2.984 1 2 3 4 5 6 7 8 9 170.5 187 191.5 195 198.5 202 205.5 209 213.5 220 187 1442 1546 1593 1669 1436 1619 1609 1553 1701 1581 15749 0.0916 0.0982 0.1011 0.1060 0.0912 0.1028 0.1022 0.0986 0.1080 0.1004 1.0001 191.5 195 198.5 202 205.5 209 213.5 220 245.5 10 Sum N N ( ) 2 m X X X i i X E X V X ( ) 2 m = = s = = m = 2 X = = 1 N 1 i i E X X X N ( ) 2 2 m = + m m = m 2 2 X 2 Note (for later): 2 E X E X X E X X X
Bivariate Association Between Over/Under (X) and Total Points (Y) - I N N ( ) 2 m X X i i X m = = m = s = = s = s = s = 2 X 2 Y = = Means: 202.817 203.213 Variances/SDs: 156.235 12.499 458.054 21.402 1 N 1 i i X Y X Y N N ( )( ) m m X Y i X i Y ( )( ) ( )( ) = m m = s = = m m = m m = Covariance: COV , 151.972 Note: 1 i X Y E X Y E X Y E XY X Y XY X Y X Y N s s s = = = Correlation: CORR , 0.568 X Y XY XY X Y s s m = + = = = m m Conditional Mean: with 5.930 X XY | 0 1 1 0 1 Y X Y X 2 X s s s s s s s s s s s s s s s s s ( ) ( ) ( ) ( ) m = m m + = m + m = m + m = m + m = m + m X X X X X XY XY XY XY Y XY Y Y | Y X Y X Y X Y X Y X Y XY X 2 X 2 X 2 X 2 X Y X Y X X 21.402 12.499 ( ) m = + = 5.930 0.973 + 203.213 0.568 202.817 X X | Y X ( ) ( ) s = s = = s = 2 Y X 2 Y 2 XY 2 2 Conditional Variance/Standard Deviation: 1 21.402 1 .568 310.269 17.614 | | Y X
Bivariate Association Between Over/Under (X) and Total Points (Y) - II
Conditional Distributions of X and Y Given Z - I z E{X|z} E{Y|z} V{X|z} V{Y|z} Cov{X,Y|z} Cor{X,Y|z} N(z) 9.644 1.058 0.921 0.918 0.884 1.764 0.668 1.438 2.230 23.496 p(z) 1 2 3 4 5 6 7 8 9 182.982 189.104 193.020 196.453 199.986 203.524 207.005 210.852 216.181 226.220 184.272 190.633 193.653 197.003 200.055 202.853 207.002 210.612 216.609 226.675 9.138 1.738 0.973 1.002 1.027 0.999 1.009 1.682 3.288 24.996 293.998 291.372 274.834 290.976 320.250 315.420 312.277 337.123 308.917 394.628 0.186 0.047 0.056 0.054 0.049 0.099 0.038 0.060 0.070 0.237 1442 1546 1593 1669 1436 1619 1609 1553 1701 1581 0.0916 0.0982 0.1011 0.1060 0.0912 0.1028 0.1022 0.0986 0.1080 0.1004 10 Note: 1) The conditional means of X (Over/Under) and Y (Total Points) are very similar with each level of z (corr = .999) 2) The conditional means of X and Y are increasing in z (which is a partition of the Over/Under range) not surprising 3) The conditional correlations between X and Y are much smaller within z levels than overall ( XY = 0.568) 4) The highest conditional correlations are in the two extreme z levels (low score and high score games)
Conditional Expectations, Variances, and Covariances - I Notation (Discrete Random Variables , Y, , generalizes to Continuous RVs): X Z ( ) , p x z p ( ) ( ) z ( ) ( ) ( ) z = = = = = = = = = = , , , | | z s.t. 0 P X x Z z p x z P Z z p p x z P X x Z z p x z p ( ) z | Z X Z Z x Z Conditional Expectation: | E X Z z = ( ) ( ) z = | z s.t. 0 | is random variable wrt xp x z Z p E X Z Z | X Z x Conditional Variance: 2 2 = = 2 | | | | | V X Z E X E X Z Z E X Z E X Z Conditional Covariance: = = COV , | | | | | | | X Y Z E X E X Z Y E Y Z Z E XY Z E X Z E Y Z z E{X|z} E{Y|z} V{X|z} V{Y|z} Cov{X,Y|z} Cor{X,Y|z} N(z) 9.644 1.058 0.921 0.918 0.884 1.764 0.668 1.438 2.230 23.496 p(z) 1 2 3 4 5 6 7 8 9 182.982 189.104 193.020 196.453 199.986 203.524 207.005 210.852 216.181 226.220 184.272 190.633 193.653 197.003 200.055 202.853 207.002 210.612 216.609 226.675 9.138 1.738 0.973 1.002 1.027 0.999 1.009 1.682 3.288 24.996 293.998 291.372 274.834 290.976 320.250 315.420 312.277 337.123 308.917 394.628 0.186 0.047 0.056 0.054 0.049 0.099 0.038 0.060 0.070 0.237 1442 1546 1593 1669 1436 1619 1609 1553 1701 1581 0.0916 0.0982 0.1011 0.1060 0.0912 0.1028 0.1022 0.0986 0.1080 0.1004 10
Conditional Expectations, Variances, and Covariances - II Conditional Expectations to Marginal Expectations : | | z s.t. 0 X Z Z x E X E E X Z E X Z z p z = = = ( ) ( ) z = = | is random variable wrt E X Z z xp x z p E X Z Z | ( ) | | Z z Conditional Variances and Expectations to Marginal Varia | | V E X Z E E X Z = nce: 2 2 = = 2 | | | | | V X Z E X E X Z Z E X Z E X Z 2 2 = = 2 2 | | | | E V X Z E E X Z E E X Z E X E E X Z 2 E X 2 2 2 = | | E E X Z E E X Z E X E X V X 2 2 2 2 + = + = = 2 2 | | | | E V X Z V E X Z E X E E X Z E E X Z E X Co nditional Covariances and Expectations to Marginal Covariance: E X E Y = = COV , | | | | | | | X Y Z E X E X Z Y E Y Z Z E XY Z E X Z E Y Z , X Y = = COV , | | | | | | E X Y Z E E XY Z E E X Z E Y Z E XY E E X Z E Y Z , E Y Z E X E Y = = COV | , | | | | | | | E X Z COV E Y Z E E X Z E Y Z E E X Z E E Y Z E E X Z E Y Z + = = , | COV | | COV E X Y Z E X Z E XY 1) Marginal Mean = Probability weighted average of conditional means 2) Marginal Variance = Mean(Conditional Variances) + Variance(Conditional Means) 3) Marginal Covariance = Mean(Conditional Covariances) + Covariance(Conditional Means)
Application: NBA Over/Under (X), Total Points (Y), OU Group (Z) z E{X|z} E{Y|z} V{X|z} V{Y|z} Cov{X,Y|z} Cor{X,Y|z} N(z) 9.644 1.058 0.921 0.918 0.884 1.764 0.668 1.438 2.230 23.496 p(z) 1 2 3 4 5 6 7 8 9 182.982 189.104 193.020 196.453 199.986 203.524 207.005 210.852 216.181 226.220 184.272 190.633 193.653 197.003 200.055 202.853 207.002 210.612 216.609 226.675 9.138 1.738 0.973 1.002 1.027 0.999 1.009 1.682 3.288 24.996 293.998 291.372 274.834 290.976 320.250 315.420 312.277 337.123 308.917 394.628 0.186 0.047 0.056 0.054 0.049 0.099 0.038 0.060 0.070 0.237 1442 1546 1593 1669 1436 1619 1609 1553 1701 1581 0.0916 0.0982 0.1011 0.1060 0.0912 0.1028 0.1022 0.0986 0.1080 0.1004 10 10 ( ) z E X m = = = 182.982(.0916) ... 226.220(.1004) + + = m = | 202.817 203.213 E X z p X Z Y = 1 10 z ( ) z = = 9.138(.0916) ... 24.996(.1004) + + = = | | 4.539 | 313.711 E V X Z V X z p E V Y z Z = 1 z 10 E X ( ) z ( ) ( ) 2 2 2 = = 182.982 202.817 + + 220 202.817 = = | | (.0916) ... 226. (.1004) 151.693 | 144.143 V E X Z E X z p V E Y z Z = 1 z 10 ( ) z = = 9.644(.0916) ... 23.496(.1004) + + = COV , | COV , | 4.246 E X Y Z X Y z p Z = 1 z 10 E X E Y ( ) z = = COV | , | | | E X Z E Y Z E X z E Y z p Z = 1 z ( )( ) ( )( = ) = 182.982 202.817 184.272 203.213 (.0916) ... 4.539 151.693 156.232 V X = = + + V Y + 220 202.817 226.675 203.213 (.1004) = 226. 147.723 s = s = = 313.711 144.143 + s = = 4.246 147.723 151.969 + = 2 X 2 Y 457.854 COV , X Y XY
