The accompanying table shows, for credit-card holders with...

The accompanying table shows, for credit-card holders with one to three cards, the joint probabilities for number of cards owned $(X)$ and number of credit purchases made in a week $(Y)$. $$ \begin{array}{cccccc} \hline \begin{array}{c} \text { Number of } \\ \text { Cards }(X) \end{array} & \underline{\text { Number of Purchases in Week }(n)} \\ & 0 & 1 & 2 & 3 & 4 \\ \hline 1 & 0.08 & 0.13 & 0.09 & 0.06 & 0.03 \\ 2 & 0.03 & 0.08 & 0.08 & 0.09 & 0.07 \\ 3 & 0.01 & 0.03 & 0.06 & 0.08 & 0.08 \\ \hline \end{array} $$ a. For a randomly chosen person from this group, what is the probability distribution for number of purchases made in a week? b. For a person in this group who has three cards, what is the probability distribution for number of purchases made in a week? c. Are number of cards owned and number of purchases made statistically independent?

The accompanying table shows, for credit-card holders with one to three cards, the joint probabilities for number of cards owned $(X)$ and number of credit purchases made in a week $(Y)$.
$$
\begin{array}{cccccc}
\hline \begin{array}{c}
\text { Number of } \\
\text { Cards }(X)
\end{array} & \underline{\text { Number of Purchases in Week }(n)} \\
& 0 & 1 & 2 & 3 & 4 \\
\hline 1 & 0.08 & 0.13 & 0.09 & 0.06 & 0.03 \\
2 & 0.03 & 0.08 & 0.08 & 0.09 & 0.07 \\
3 & 0.01 & 0.03 & 0.06 & 0.08 & 0.08 \\
\hline
\end{array}
$$
a. For a randomly chosen person from this group, what is the probability distribution for number of purchases made in a week?
b. For a person in this group who has three cards, what is the probability distribution for number of purchases made in a week?
c. Are number of cards owned and number of purchases made statistically independent?

Key Concepts

Joint Probability Distribution

A joint probability distribution describes the probability of two (or more) random variables occurring simultaneously. It provides a complete picture of the relationship between the variables by assigning probabilities to each pair (or tuple) of outcomes in their respective sample spaces.

Marginal Probability Distribution

A marginal probability distribution is obtained by summing or integrating the joint probability distribution over the unwanted variable(s). It represents the probabilities of a single variable irrespective of the values of the other variable(s) and is useful for understanding the distribution of individual random variables in a joint context.

Conditional Probability Distribution

A conditional probability distribution describes the probability of one random variable given the occurrence of a specific value or condition of another random variable. It is derived from the joint probability distribution and helps to isolate the behavior of one variable while considering the effect of another.

Statistical Independence

Statistical independence between two random variables means that the occurrence of one does not affect the probability of occurrence of the other. Mathematically, this is usually expressed as the joint probability being equal to the product of the marginal probabilities for all combinations of the variables.

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