Below is the solution to the first part of an equation, I need to know the detai

Question

Below is the solution to the first part of an equation, I need to know the detailed explanation of how 683.905 was found. (it was found from the similar table as presented above but with different figures)

Age and Hourly Earnin 5 0.001431 0.001943 0.001636 6 0.001431 0.000613 0.001534 0.001431| 0.001943| 0.000818 8 0.002351 0.001534 0.001738 9 0.002351 0.002965 0.002045 100.004192 0.004805 0.003681 11 0.003272 0.003476 0.002863 12 0.00685 0.006134 0.005623 13 0.003885 0.003681 0.005112 140.0058280.004907 0.007054 15 0.003476 0.003169 0.00409 16 0.002658 0.003272 0.004192 17 0.004703 0.00501 0.006441 18 0.002658 0.003169 0.003578 19 0.003681 0.00409 0.005214 20 0.002965 0.002863 0.003067 22 0.003885 0.004601 0.005419 24 0.004090.005725 0.006748 26 0.0031690.003169 0.00501 28 0.002965 0.002658 0.003681 30 0.001943 0.002249 0.00409 0.002556 0.002658 0.004907 400.0013290.002249 0.001636 450.001022 0.000511 0.001431 500.000511 0.000511 0.001227 55 0.000102 0.000818 0.000613 0.000204 0.000613 0.000307 65 0.000102 0.000204 0.000102 70 0.000307 0.000204 0.000818 0.001942542 0.001431346 0.001533586 0.00102239 0.00102239 0.001022 0.001125 0.001227 0.000511 0.001125 0.0011246290.001227 0.001431 0.000409 0.001329 0.00040 0.0009201510.001738 0.001022 0.00092 0.000613 0.00112 0.0013291070.000818 0.001329 0.00184| 0.001227 0.001329 0.0022492590.001943 0.001636 0.001227 0.001636 0.001227 0.0036806050.004192 0.003578 0.002965 0.00276 0.002147 0.0024537370.003067 0.002045 0.002863 0.002965 0.002249 0.0058276250.005521 0.004805 0.00501 0.00501 0.004601 0.003578366 0.003987 0.003067 0.003272 0.003681 0.003374 0.0047029960.006646 0.006748 0.005828 0.004907 0.005419 0.003987322 0.005112 0.0039870.00501 0.002351 0.0035 0.0048052350.003272 0.004396 0.003681 0.003169 0.002147 0.0058276250.004703 0.005828 0.0070540.004805 0.005214 0.00316941 0.003681 0.003681 0.003476 0.003681 0.00378 0.0060321030.006646 0.006339 0.005828 0.00685 0.004601 0.0036806050.003885 0.004192 0.00501 0.005419 0.003374 0.0070544930.007975 0.006646 0.007975 0.007361 0.007054 0.0076679280.007872 0.011349 0.007566 0.009713 0.008997 0.0044985180.005419 0.004499 0.005316 0.004601 0.005828 0.004805235 0.00685 0.005725 0.006543 0.006237 0.007872 0.0052 0.005725 0.006032 0.006032 0.006237 0.0062365810.007054 0.006237 0.008281 0.010326 0.00685 0.0030671710.004294 0.004396 0.0061340.004703 0.005214 0.001738064 0.003272 0.003885 0.003783 0.005214 0.004601 0.0012268680.001738 0.00276 0.002249 0.002556 0.003067 0.004294039 0.004396278 0.003885083 0.006134342 0.003578366 0.003067171 0.00633882 0.002964932 0.004805235 0.002760454 0.006543298 0.00950823 0.003476127 0.004600757 0.003476127 0.006952254 0.002862693 0.001533586 0.001226868 0.000817912 0.000511195 0.000102239 0.000715673 214191 0.005316 511195 0.001022 0.000818 0.001125 0.001534 0.001738 15673 0.000818 0.00071 0.001125 0.00092 0.001738 0.001329 0.0005111950.000204 0.000409 0.000204 0.000716 0.000818 0.000613434 0.001227 0.002147 0.001636 0.001738 0.003169

Explanation / Answer

The first part of the covariance can be calculated by multiplying each value of the probability in the table with the corresponding values of age and hourly earnings, i.e. by multiplying probability in each cell by its corresponding column header value (age) and row header value (hourly earnings). The sum of these will give the desired value.

In the sample given, the first probability of 0.001431 has age 25 and hourly earnings of 5. We multiply these = 0.001431*25*5 = 0.178875. We repeat this for all the probability cells. Once we have these values, we sum them up to get the desired value.

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