A market researcher wants to determine whether a new model...

A market researcher wants to determine whether a new model of a personal computer that had been advertised on a late-night talk show had achieved more brand-name recognition among people who watched the show regularly than among people who did not. After conducting a survey, it was found that $15 \%$ of all people both watched the show regularly and could correctly identify the product. Also, $16 \%$ of all people regularly watched the show and $45 \%$ of all people could correctly identify the product. Define a pair of random variables as follows: $$ \begin{array}{llll} \hline X=1 & \text { if regularly watch the show } & X=0 & \text { otherwise } \\ \hline Y=1 & \text { if product correctly identified } & Y=0 & \text { otherwise } \\ \hline \end{array} $$ a. Find the joint probability distribution of $X$ and $Y$. b. Find the conditional probability distribution of $Y$, given $X=1$. c. Find and interpret the covariance between $X$ and $Y$.

A market researcher wants to determine whether a new model of a personal computer that had been advertised on a late-night talk show had achieved more brand-name recognition among people who watched the show regularly than among people who did not. After conducting a survey, it was found that $15 \%$ of all people both watched the show regularly and could correctly identify the product. Also, $16 \%$ of all people regularly watched the show and $45 \%$ of all people could correctly identify the product. Define a pair of random variables as follows:
$$
\begin{array}{llll}
\hline X=1 & \text { if regularly watch the show } & X=0 & \text { otherwise } \\
\hline Y=1 & \text { if product correctly identified } & Y=0 & \text { otherwise } \\
\hline
\end{array}
$$
a. Find the joint probability distribution of $X$ and $Y$.
b. Find the conditional probability distribution of $Y$, given $X=1$.
c. Find and interpret the covariance between $X$ and $Y$.

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Key Concepts

Joint Probability Distribution

The joint probability distribution describes the probability that two or more random variables take on specific values simultaneously. It provides a complete description of the relationship between the variables and allows for the computation of marginal probabilities, which are obtained by summing over specific variables, as well as conditional probabilities.

Conditional Probability Distribution

The conditional probability distribution gives the probabilities of outcomes of a random variable given that another variable has already taken a specific value. It helps in understanding how the probability of one event is affected by the occurrence of another event, making it a key tool in the analysis of dependent random variables.

Covariance

Covariance is a measure of the extent to which two random variables change together. It quantifies the joint variability of the variables; a positive covariance indicates that the variables tend to increase together, while a negative covariance suggests that as one variable increases, the other tends to decrease. This concept is fundamental in identifying and interpreting linear relationships between variables in probability and statistics.

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