Students in a college were classified according to years in...

Students in a college were classified according to years in school $(X)$ and number of visits to a museum in the last year $(Y=0$ for no visits, 1 for one visit, 2 for more than one visit). The joint probabilities in the accompanying table were estimated for these random variables. $$ \begin{array}{ccccc} \hline {\begin{array}{c} \text { Number of } \\ \text { Visits }(Y) \end{array}} & \underline {\text { Years in School }(X)} \\ & 1 & 2 & 3 & 4 \\ \hline 0 & 0.07 & 0.05 & 0.03 & 0.02 \\ 1 & 0.13 & 0.11 & 0.17 & 0.15 \\ 2 & 0.04 & 0.04 & 0.09 & 0.10 \\ \hline \end{array} $$ a. Find the probability that a randomly chosen student has not visited a museum in the last year. b. Find the means of the random variables $X$ and $Y$. c. Find and interpret the covariance between the random variables $X$ and $Y$.

Students in a college were classified according to years in school $(X)$ and number of visits to a museum in the last year $(Y=0$ for no visits, 1 for one visit, 2 for more than one visit). The joint probabilities in the accompanying table were estimated for these random variables.
$$
\begin{array}{ccccc}
\hline {\begin{array}{c}
\text { Number of } \\
\text { Visits }(Y)
\end{array}} & \underline {\text { Years in School }(X)} \\
& 1 & 2 & 3 & 4 \\
\hline 0 & 0.07 & 0.05 & 0.03 & 0.02 \\
1 & 0.13 & 0.11 & 0.17 & 0.15 \\
2 & 0.04 & 0.04 & 0.09 & 0.10 \\
\hline
\end{array}
$$
a. Find the probability that a randomly chosen student has not visited a museum in the last year.
b. Find the means of the random variables $X$ and $Y$.
c. Find and interpret the covariance between the random variables $X$ and $Y$.

Key Concepts

Joint Probability Distribution

A joint probability distribution specifies the probability that each possible pair of outcomes for two random variables occurs simultaneously. It is usually presented as a table, function, or formula, and it is the foundation for computing probabilities, marginal distributions, and expectations when dealing with two variables at the same time.

Marginal Distribution

A marginal distribution is derived from the joint distribution by summing the probabilities over one variable to get the distribution for the other variable. This step is crucial for analyzing each random variable independently, such as finding the probability of a single event irrespective of the other variable.

Expected Value

The expected value, or mean, of a random variable represents the long-run average outcome if an experiment were repeated many times. For a discrete random variable, it is calculated as the sum of the product of each outcome value and its corresponding probability. This concept helps summarize the central tendency of the distribution.

Covariance

Covariance is a measure of the degree to which two random variables change together. It is computed as the expected value of the product of their deviations from their respective means, formally E[XY] – E[X]E[Y]. A positive covariance indicates that the variables tend to increase together while a negative covariance suggests that as one increases the other tends to decrease. It helps understand the directional relationship between the variables.

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