Question

A candy company distributes boxes of chocolates with a mixture of creams, toffees, and nuts coated in both light and dark chocolate. For a randomly selected box, let X and Y be the proportion of the light and respectively dark chocolates that are creams and suppose the joint density function is given by: f_{(X,Y)}(x,y)=egin{cases} frac{5}{2}(2x+3y), & 0le xle 1, 0le yle 1 \ 0, & ext{otherwise} end{cases} (a) (10 points) Find P((X,Y)in A) where A is the region {(x,y): 0 < x < 1/2, 1/4 < y < 1/2}; (b) (10 points) Find the probability density functions of the random variables X and Y; (c) (20 points) Find E(X), Var(X), E(Y), Var(Y), E(2X - Y) and Var(2X - Y); (d) (10 points) Find E(XY), covariance Cov(X,Y) and correlation coefficient ?(X,Y).

          A candy company distributes boxes of chocolates with a mixture of creams, toffees, and nuts coated in both light and dark chocolate. For a randomly selected box, let X and Y be the proportion of the light and respectively dark chocolates that are creams and suppose the joint density function is given by:

f_{(X,Y)}(x,y)=egin{cases} 	frac{5}{2}(2x+3y), & 0le xle 1, 0le yle 1 \ 0, & 	ext{otherwise} end{cases}

(a) (10 points) Find P((X,Y)in A) where A is the region {(x,y): 0 < x < 1/2, 1/4 < y < 1/2};
(b) (10 points) Find the probability density functions of the random variables X and Y;
(c) (20 points) Find E(X), Var(X), E(Y), Var(Y), E(2X - Y) and Var(2X - Y);
(d) (10 points) Find E(XY), covariance Cov(X,Y) and correlation coefficient ?(X,Y).

$A candy company distributes boxes of chocolates with a mixture of creams, toffees, and nuts coated in both light and dark chocolate. For a randomly selected box, let X and Y be the proportion of the light and respectively dark chocolates that are creams and suppose the joint density function is given by: f(X,Y)(x,y)=egincases frac52(2x+3y), 0le xle 1, 0le yle 1 0, extotherwise endcases (a) (10 points) Find P((X,Y)in A) where A is the region (x,y): 0 < x < 1/2, 1/4 < y < 1/2; (b) (10 points) Find the probability density functions of the random variables X and Y; (c) (20 points) Find E(X), Var(X), E(Y), Var(Y), E(2X - Y) and Var(2X - Y); (d) (10 points) Find E(XY), covariance Cov(X,Y) and correlation coefficient ?(X,Y).$

Added by Esperanza F.

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