A privately owned liquor store operates both a drive-n facility and a walk-in facility. On a randomly selected day, let X and Y, respectively, be the proportions of the time that the drive-in and walk-in facilities are in use, and suppose that the joint density function of these variables is f(x,y)={23(x+2y)0≤x≤1, 0≤y≤10,elsewhere a) Find the marginal density of X. b) Find the marginal density of Y. c) Find the probability that the drive-in facility is busy less than one-half of the time.

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A privately owned liquor store operates both a drive-n facility and a walk-in facility. On a randomly selected day, let X and Y, respectively, be the proportions of the time that the drive-in and walk-in facilities are in use, and suppose that the joint density function of these variables is f(x,y)={23(x+2y)0≤x≤1,  0≤y≤10,elsewhere a) Find the marginal density of X. b) Find the marginal density of Y. c) Find the probability that the drive-in facility is busy less than one-half of the time.

ossidianaZ · Accepted Answer

Step 1 a) The marginal density of X is: f(x)=∫01f(x,y)dy =∫0123(x+2y)dy =2x3∫01dy+43∫01ydy =2x3[y]01+43[y22]01 =2x2+23 =2(x+1)3, 0≤x≤1 Step 2 b) The marginal density of Y is given below: f(y)=∫01f(x,y)dx =∫0123(x+2y)dx =23∫01xdx+4y3∫01dx =23[x22]01+4y3[x]01 13+4y3 =1+4y3, 0≤y≤1 c) It is given that drive in facility is denoted by x, then P(X&amp;lt;12)=∫012f(x)dx =∫0122(x+1)3dx =23∫012(x+1)dx

A privately owned liquor store operates both a drive-n facility and a walk-in facility. On a randomly selected day, let X and Y, respectively, be the proportions of the time that the drive-in and walk-in facilities are in use, and suppose that the joint density function of these variables is.

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