A real estate agent is interested in the relationship...

A real estate agent is interested in the relationship between the number of lines in a newspaper advertisement for an apartment and the volume of inquiries from potential renters. Let volume of inquiries be denoted by the random variable $X$, with the value 0 for little interest, 1 for moderate interest, and 2 for strong interest. The real estate agent used historical records to compute the joint probability distribution shown in the accompanying table. $$ \begin{array}{cccc} \hline {\begin{array}{c} \text { Number } \\ \text { of Lines }(n) \end{array}} & \underline {\text { Number of Inquiries }(X)} \\ & 0 & 1 & 2 \\ \hline 3 & 0.09 & 0.14 & 0.07 \\ 4 & 0.07 & 0.23 & 0.16 \\ 5 & 0.03 & 0.10 & 0.11 \\ \hline \end{array} $$ a. Find the joint cumulative probability at $X=1, Y=4$, and interpret your result. b. Find and interpret the conditional probability distribution for $Y$, given $X=0$. c. Find and interpret the conditional probability distribution for $X$, given $Y=4$. d. Find and interpret the covariance between $X$ and $Y$. e. Are number of lines in the advertisement and volume of inquiries independent of one another?

A real estate agent is interested in the relationship between the number of lines in a newspaper advertisement for an apartment and the volume of inquiries from potential renters. Let volume of inquiries be denoted by the random variable $X$, with the value 0 for little interest, 1 for moderate interest, and 2 for strong interest. The real estate agent used historical records to compute the joint probability distribution shown in the accompanying table.
$$
\begin{array}{cccc}
\hline {\begin{array}{c}
\text { Number } \\
\text { of Lines }(n)
\end{array}} & \underline {\text { Number of Inquiries }(X)} \\
& 0 & 1 & 2 \\
\hline 3 & 0.09 & 0.14 & 0.07 \\
4 & 0.07 & 0.23 & 0.16 \\
5 & 0.03 & 0.10 & 0.11 \\
\hline
\end{array}
$$
a. Find the joint cumulative probability at $X=1, Y=4$, and interpret your result.
b. Find and interpret the conditional probability distribution for $Y$, given $X=0$.
c. Find and interpret the conditional probability distribution for $X$, given $Y=4$.
d. Find and interpret the covariance between $X$ and $Y$.
e. Are number of lines in the advertisement and volume of inquiries independent of one another?

Key Concepts

Joint Probability Distribution

A joint probability distribution describes the probability of each combination of outcomes for two discrete random variables. It provides a complete picture of the likelihood for every pair of values that the variables can take, allowing for further analysis such as computation of marginal and conditional probabilities.

Joint Cumulative Distribution Function (CDF)

The joint cumulative distribution function for two random variables gives the probability that each variable takes on a value less than or equal to specified thresholds. It aggregates probabilities from the joint distribution, providing insight into the accumulation of probability over a region in the sample space.

Conditional Probability Distribution

A conditional probability distribution specifies the probabilities of outcomes for one random variable given that another variable is fixed at a specific value. This concept is crucial for understanding how one variable behaves in light of information about the other, and it is calculated by dividing the joint probability by the marginal probability of the given condition.

Covariance

Covariance is a measure that indicates the extent to which two random variables change together. A positive covariance means that as one variable increases, the other tends to increase as well, whereas a negative covariance indicates an inverse relationship. It is fundamental in assessing the strength and direction of any linear relationship between the variables.

Independence of Random Variables

Two random variables are considered independent if the occurrence of one does not affect the probability of occurrence of the other. In the context of a joint distribution, independence is verified by checking if the joint probability for any pair of outcomes equals the product of their respective marginal probabilities.

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