Question

A​ fast-food restaurant operates both a​ drive-through 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-through and​ walk-in facilities are in​ use, and suppose that the joint density function of these random variables is as shown below. Complete parts​ (a) through​ (c).

​f(x,y)equals=

left brace Start 2 By 2 Matrix 1st Row 1st Column two sevenths left parenthesis 5 x plus 2 y right parenthesis comma 2nd Column 0 less than or equals x less than or equals 1 comma 0 less than or equals y less than or equals 1 2nd Row 1st Column 0 comma 2nd Column elsewhere EndMatrix 2/7(5x+2y), 0≤x≤1, 0≤y≤1 0, elsewhere

​(a) Find the marginal density of X.

Select the correct choice below and fill in the answer box to complete your choice.

A.

​h(y)equals=nothing​,

for

0less than or equals≤yless than or equals≤1

B.

​g(x)equals=nothing​,

for

0less than or equals≤xless than or equals≤1

Answer 1

a)

marginal density of X =g(x)= f(x,y)dy = (2/7)*(5x+2y) dy

g(x) =(2/7)*(5xy+2y²/2) |¹₀

g(x)=(2/7)*(5x+1)    for 0 <x<1