Question

3.40 A fast-food restaurant operates both a drivethrough 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 f(x, y) = 2 3 (x + 2y), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0, elsewhere. (a) Find the marginal density of X. (b) Find the marginal density of Y . (c) Find the probability that the drive-through facility is busy less than one-half of the time.

Answer 1

Answer:

a) Marginal probability of X

$f_x(x)=\frac{2}{3}(x+1)$

b) Marginal probability of Y

$f_y(y)=\frac{2}{3}(1+2y)$

c) The probability that the drive-through facility is busy less than one-half of the time is P=0.5.

Step-by-step explanation:

The joint density function of probability is:

$f(x, y) = (2/3) (x + 2y)\\\\0 \leq x \leq 1, \,0 \leq y \leq 1$

a) The marginal density of X is

$f_x(x)=\int_0^1 f(x,y)dy=(2/3)(x+2\int_0^1 y\,dy)=(2/3)*(x+y^2)\\\\f_x(x)=(2/3)*(x+(1-0))\\\\f_x(x)=(2/3)*(x+1)$

b) The marginal density of Y is:

$f_y(y)=\int_0^1 f(x,y)dx=(2/3)*(\int_0^1x\,dx+2y)=(2/3)*(1^2-0^2+2y)\\\\f_y(y)=(2/3)*(1+2y)$

c) Probability that the drive-through facility is busy less than one-half of the time. This is P(X<0.5).

$P(X$