The value of E[X Y] is 44.25.
According to the question,
E[X] = ∑ x × P(X=x)
=0×P(x=0) + 5×P(x=1) + 10×P(x=2)
= 0+5×(0.49)+10×(0.31)
= 5.55
E[Y] = ∑ y × P(Y=y)
=0×P(x=0) + 5×P(x=5) + 10×P(x=10) + 15×P(x=15)
= 0+5×(0.49)+10×(0.31)+15×(0.21)
= 8.55
Now,
E [X²] = ∑ x² × P(X=x)
= 0²×P(x=0) + 5²×P(x=1) + 10²×P(x=2)
= 0 + 25×0.49 + 100×0.31
= 43.25
E[Y²] = ∑ y² × P(Y=y)
= 0²×P(x=0) + 5²×P(x=5) + 10²×P(x=10) + 15²×P(x=15)
= 0 + 25×0.49 + 100×0.31 + 225×0.21
= 92.25
Therefore , V[X] = E[X²] - {E[X]} = 43.25 - 30.8025 =12.4475
V[Y] = E[Y²] - {E[Y]}² = 92.25 - 73.1025 = 19.1475
E[X Y] = ∑xy P(X=x, Y=y)
= {0×0×P(X=0,Y=0)} + {0×5×P(X=0,Y=5)} + {0×10×P(X=0,Y=0)} + {0×15×P(X=0, Y =15)} + {5×0×P(X=5,Y=0)} + {5×10×P(x=5,Y=10)} + {5×5×P(x=5,Y=5)} + {5×15×P(x=5,Y=15)} + {10×0×P(x=10,Y=0)} + {10×5×P(x=10,Y=5)} + {10×10×P(x=10,Y=10)} + {10×15×P(x=10,Y=15)}
= 0+0+0+0+0 + {50×0.20} +{25×0.15} + {75×0.10} + 0 + {50×0.15} + {100×0.14} + {150×0.01}
= 10 + 3.75 + 7.5 + 7.5 + 14 + 1.5
Hence the final answer is = 41.25
To learn more about probability and their theorems,
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