Problem 4) A company distributes boxes of three products: Product 1, Product 2,

Question

Problem 4) A company distributes boxes of three products: Product 1, Product 2, and Product 3. Suppose that the weight of each box is 1 kilogram, but the individual weights ofproducts 1, 2 and 3 vary from box to box. For a randomly selected box, let X and Y represent the weights of Product 1 and Product 2, respectively. Suppose that the joint density function of X and Y is 1x, y) _{24ry, Otherwise.OSYS!, x + y S 1. a) Find the probability that in a given box Product 3 accounts for more than 1/2 of the weight. Find the marginal density for the weight of Product l kilogram if it is known that Product 1 constitute 3/4 ofthe weight. 2 in these boxes. purchased a box of products. b) c) Find the probability that the weight of Product 2 in a box is less than 1/8 of a d) Find the covariance between the weight of Product 1 and the weight of Product e) Find the expected weight for the sum of Product 1 and Product 2 if one

Explanation / Answer

a) The probability for x+y<1/2 is P(y<1/2-x)= 24xyd(x)d(y) y-(0,1/2-x) and x(0,1) thus it is 12x*(1/2-x)^2d(x)

or 12*x*(1/4+x^2-x)=12*(x/4+x^3-x^2)d(x) solving we get 12*(x^2/8+x^4/4-x^3/3) putting limits we hget the probability as 12*(1/8+1/4-1/3)=12*(1/8-1/12)=1/2.

b) Marginal density of x is P(y<x-1)= 24xyd(y) y-(0,1-x) or 12x*(1-x)^2 for x[0,1].

c) Marginal density of y is P(x<y-1) similarly as above it can be solved to be 12y*(1-y)^2 for y-[0,1]

thus P(y<1/8) becomes 12*(y+y^3-2y^2)dy from [0,1/8) or 12*(y^2/2+y^4/4-2/3y^3) putting y=1/8

we get the probability as 12*(1/128+1/128*1/128*1/4-1/12*1/64)=10/6*1/128+3/128*1/128=.01

d)The mean for x and y can be found as 12x^2*(1-x)^2dx from-[0,1] which is 12*(x^2+x^4-2x^3)dx

solving we get 12*(1/3+1/5-2/4)=12*(1/5-1/6)=12/30=0.4. for both x and y.

Covariance thus becomes 24xy*(x-0.4)*(y-0.4)dxdy y-[0,1-x] and x[0,1].

e)E(x+y)=24xy*(x+y)dxdy y-[0,1-x] and x-[0,1] solving the integration we get 24x*(x*y^2/2+y^3/3)dx putting limits we get 24x(x(1-x)^2/2+(1-x)^3/3)dx from [0,1]

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