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.

UkusakazaL

UkusakazaL

Answered question

2021-07-31

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)0x1,  0y10,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.

Answer & Explanation

ossidianaZ

ossidianaZ

Beginner2021-08-02Added 2 answers

Step 1
a) The marginal density of X is:
f(x)=01f(x,y)dy
=0123(x+2y)dy
=2x301dy+4301ydy
=2x3[y]01+43[y22]01
=2x2+23
=2(x+1)3, 0x1
Step 2
b) The marginal density of Y is given below:
f(y)=01f(x,y)dx
=0123(x+2y)dx
=2301xdx+4y301dx
=23[x22]01+4y3[x]01
13+4y3
=1+4y3, 0y1
c) It is given that drive in facility is denoted by x, then
P(X<12)=012f(x)dx
=0122(x+1)3dx
=23012(x+1)dx

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