A college bookseller makes calls at the offices of...

A college bookseller makes calls at the offices of professors and forms the impression that professors are more likely to be away from their offices on Friday than any other working day. A review of the records of calls, $1 / 5$ of which are on Fridays, indicates that for $16 \%$ of Friday calls, the professor is away from the office, while this occurs for only $12 \%$ of calls on every other working day. Define the random variables as follows: $$ \begin{array}{llll} \hline X=1 & \text { if call is made on a Friday } & X=0 & \text { otherwise } \\ \hline Y=1 & \text { if professor is away from the office } & Y=0 & \text { otherwise } \\ \hline \end{array} $$ a. Find the joint probability distribution of $X$ and $Y$. b. Find the conditional probability distribution of $Y$, given $X=0$. c. Find the marginal probability distributions of $X$ and $Y$. d. Find and interpret the covariance between $X$ and $Y$.

A college bookseller makes calls at the offices of professors and forms the impression that professors are more likely to be away from their offices on Friday than any other working day. A review of the records of calls, $1 / 5$ of which are on Fridays, indicates that for $16 \%$ of Friday calls, the professor is away from the office, while this occurs for only $12 \%$ of calls on every other working day. Define the random variables as follows:
$$
\begin{array}{llll}
\hline X=1 & \text { if call is made on a Friday } & X=0 & \text { otherwise } \\
\hline Y=1 & \text { if professor is away from the office } & Y=0 & \text { otherwise } \\
\hline
\end{array}
$$
a. Find the joint probability distribution of $X$ and $Y$.
b. Find the conditional probability distribution of $Y$, given $X=0$.
c. Find the marginal probability distributions of $X$ and $Y$.
d. Find and interpret the covariance between $X$ and $Y$.

Key Concepts

Joint Probability Distribution

A joint probability distribution specifies the probability that two (or more) random variables take on specific values simultaneously. It is represented in a table or a function form that lists all possible combinations of the outcomes of each random variable along with their corresponding probabilities. Understanding the joint distribution is crucial in analyzing how random variables interact with each other and form the foundation for calculating marginal and conditional probabilities.

Conditional Probability Distribution

A conditional probability distribution describes the probabilities of one random variable under the condition that another random variable is fixed at a certain value. It helps isolate the behavior of one variable while incorporating known information about another, and is calculated by dividing the joint probability by the marginal probability of the conditioning variable. This concept is essential for understanding the dependency structure between events or variables.

Marginal Probability Distribution

A marginal probability distribution represents the probabilities of a single random variable irrespective of the values of other variables. It is obtained by summing (or integrating) the joint probabilities over the range of the other random variables. Marginal distributions provide a summarized view of the likelihood of each outcome for an individual variable and are key in understanding the overall behavior of a subset of variables within a larger probabilistic framework.

Covariance

Covariance measures the degree to which two random variables change together. A positive covariance indicates that the variables tend to increase or decrease together, while a negative covariance suggests that as one increases, the other tends to decrease. It serves as an indicator of the linear relationship between the variables, and its computation involves the expected value of the product of the deviations of each variable from their respective means.

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