A fast-food restaurant operates both a drivethrough facility...

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 $$ f(x, y)=\left\{\begin{array}{ll} \frac{2}{3}(x+2 y), & 0 \leq x \leq 1,0 \leq y \leq 1, \\ 0, & \text { elsewhere. } \end{array}\right. $$ (a) Find the marginal density of $X$. (b) Find the marginal density of $Y$. (c) Find the probability that the drive-through facility is busy less than one-half of the time.

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
$$
f(x, y)=\left\{\begin{array}{ll}
\frac{2}{3}(x+2 y), & 0 \leq x \leq 1,0 \leq y \leq 1, \\
0, & \text { elsewhere. }
\end{array}\right.
$$
(a) Find the marginal density of $X$.
(b) Find the marginal density of $Y$.
(c) Find the probability that the drive-through facility is busy less than one-half of the time.

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Key Concepts

Joint Density Function

A joint probability density function (pdf) describes the likelihood that two or more random variables take on a specific set of values simultaneously. It is defined over a domain where the variables are valid and integrates to one over that domain. Understanding the joint pdf is crucial because it provides the basis for deriving marginal distributions and calculating probabilities for events involving multiple random variables.

Marginal Density Function

A marginal density function is obtained by integrating the joint probability density function over the range of the other variable(s). It gives the probability distribution of a single random variable irrespective of the values of the other variables. This is important for analyzing and interpreting the behavior of individual components in a multi-dimensional setting.

Calculating Probabilities via Integration

Probability calculations for events involving continuous random variables are performed by integrating the appropriate density function over the specified range. This process can involve integrating the marginal density when the event concerns a single variable, or the joint density when the event involves multiple variables. Mastering this concept is essential for solving problems in which probabilities need to be determined from given density functions.

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