If ๐X and ๐Y are independent random variables with ๐ธ[๐]=1E[X]=1, Var(๐)=2Var(X)=2, ๐ธ[๐]=3E[Y]=3, and Var(๐)Var(Y) = 4, then find ๐ธ[(2๐+3๐)2]E[(2X+3Y) 2 ].
Question
Solution 1
To find E[(2X+3Y)^2], we first need to expand the square. This gives us:
E[(2X+3Y)^2] = E[4X^2 + 12XY + 9Y^2]
Since X and Y are independent, E[XY] = E[X]E[Y]. We can substitute this into the equation:
E[(2X+3Y)^2] = E[4X^2] + E[12XY] + E[9Y^2] = 4E[X^2] + 12E[X]E[Y] + 9E[Y^2]
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