A privately owned liquor store operates both a drive-in...

A privately owned liquor store operates both a drive-in 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 random variables is $$ f(x, y)=\left\{\begin{array}{ll} \frac{2}{3}(x+2 y), & 0 \leq x \leq 1, \quad 0 \leq y \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. $$ (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.

A privately owned liquor store operates both a drive-in 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 random variables is
$$
f(x, y)=\left\{\begin{array}{ll}
\frac{2}{3}(x+2 y), & 0 \leq x \leq 1, \quad 0 \leq y \leq 1, \\
0, & \text { elsewhere. }
\end{array}\right.
$$
(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.

Key Concepts

Probability Calculation in Continuous Distributions

In continuous probability distributions, probabilities are calculated through integration of the density function over the region of interest. For instance, to determine the probability that a variable falls within a specific range, you integrate its marginal or joint density function over that interval. This concept is essential for solving problems that require finding specific probabilities, such as the likelihood of a facility being busy for a given proportion of time.

Joint Density Function

A joint density function describes the probability distribution for two continuous random variables together. It assigns a probability density to each pair of outcomes in a given two-dimensional region and must satisfy the condition that the integral over the entire space equals one. Understanding the joint density function is crucial for analyzing how the two variables interact and for calculating probabilities concerning them.

Marginal Density Function

The marginal density function represents the probability distribution of a single variable obtained by integrating the joint density function over the other variable. This operation, known as marginalization, simplifies analysis by reducing the two-dimensional distribution to a one-dimensional one, allowing you to study each variable's behavior independently.

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