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, f (x, y) ={2/3(x+2y) 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1
(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.

Step-by-step explanation:

Given

Solving (a): The marginal density of X

This is calculated as:

Integrate

Substitute 1 and 0 for y

Solving (b): The marginal density of Y

This is calculated as:

Integrate

Substitute 1 and 0 for x

Solving (c): The probability that the drive-through facility is busy less than one-half of the time.

This is represented as:

The solution is as follows:

Represent as an integral

Integrate w.r.t. x

Express as a fraction, properly

answer: ay  

fonsi  

dy  

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(sube, sube)