1. Covariance : Suppose we have two random variables and . X can take on six values (1, 2, 3, 4, 5, 6) and Y can take on seven values (1, 2, 3, 4, 5, 6, 7). Calculate the expected value, variance, standard deviation, covariance, and correlation of random variables and , using the joint probabilities below. Answers to the orange highlighted cells in the below picture.

1 0.03398716 0.01497496 0.04907765 0.00633376 0.02769399 0.0501174 0.01797557

2 0.03224175 0.02936135 0.04240043 0.00553131 0.04217874 0.043882 0.01823161

3 0.01856414 0.00927504 0.03112706 0.02130091 0.00098823 0.02132745 0.00982302

4 0.00400758 0.01368541 0.02774395 0.01481103 0.01263629 0.04788958 0.02357557

5 0.01428959 1.4051E-05 0.00570304 0.00162988 0.04057227 0.04901208 0.04319039

6 0.04461732 0.00915015 0.01149506 0.04877478 0.04966309 0.00741254 0.00373281

f(x,y) 1 2 3 4 Y 1 2 3 4 5 6 7 0.03398716 0.01497496 0.04907765 0.00633376 0.02769399 0.0501174 0.01797557 0.03224175 0.02936135 0.04240043 0.00553131 0.04217874 0.043882 0.01823161 0.01856414 0.00927504 0.03112706 0.02130091 0.00098823 0.02132745 0.00982302 0.00400758 0.01368541 0.02774395 0.01481103 0.012636290.04788958 0.02357557 0.01428959 1.4051E-05 0.00570304 0.00162988 0.04057227 0.04901208 0.04319039 0.04461732 0.00915015 0.01149506 0.04877478 0.04966309 0.00741254 0.00373281 X 5 6 Expected Value Variance S.D. Covariance Correlation X Y