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Random variables are fundamental concepts in probability theory, used to model uncertain outcomes. In this exercise, the scores given to two patients are represented as random variables, denoted as \( X_1 \) and \( X_2 \). Each variable describes possible outcomes, here being the scores \( \{1, 2, 3, 4, 5\} \).
A random variable essentially assigns a numeric value to each potential event in a sample space. A key aspect is that it can take on different values, each with a specific probability. For our patients, these probabilities are given in the problem. For example, the probability of either patient receiving a score of 1 is 0.05.
Understanding random variables is crucial as they allow analysis of probability distributions such as the probability mass function, which shows how probabilities are distributed over possible values.

Independence is a vital concept when dealing with multiple random variables. Two random variables are said to be independent if the occurrence of one does not affect the probability of the other. In the patient drug score scenario, \( X_1 \) and \( X_2 \) are considered independent.
The implication here is that the treatment effectiveness for one patient does not influence the effectiveness for the other. Therefore, calculating probabilities involving more than one random variable, such as their total or average score, can be approached by multiplying their individual probabilities. For example, to find the joint probability of one patient scoring 3 and the other scoring 4, you multiply their individual probabilities: \( P(X_1 = 3) \times P(X_2 = 4) \).
Recognizing independence simplifies many probability calculations, a key tool making it easier to understand joint distributions.

The total score is derived by summing the scores given to each patient, or \( T = X_1 + X_2 \). This represents a new random variable whose values range from 2 (\(1+1\)) to 10 (\(5+5\)). Calculating the probability mass function (PMF) for \( T \) involves finding the probability of each possible outcome.
To do this, you calculate \( P(T = t) \) for each value by summing the products of probabilities of each unique pair \( (x_1, x_2) \) that results in \( x_1 + x_2 = t \). This method allows us to comprehend how likely each total score scenario is, considering the independence of \( X_1 \) and \( X_2 \).

• For example, for \( T = 3 \), we add \( P(X_1 = 1) \cdot P(X_2 = 2) \) and \( P(X_1 = 2) \cdot P(X_2 = 1) \).

Understanding total score helps us in aggregated analysis where individual outcomes contribute towards a combined metric.

The average score is another derived metric, calculated as \( A = \frac{X_1 + X_2}{2} \). This operation results in possible values such as 1.0, 1.5, and up to 5.0. Here, the average score gives insight into the expected score per patient on average.
To find the probability of each average score, you identify combinations of \( X_1 \) and \( X_2 \) that yield the specific average and calculate the sum of their joint probabilities. Often, if the PMF of the total score \( T \) is already known, it can directly help in determining the PMF of \( A \) due to the straightforward relationship between total and average score.

• For instance, the average score of 2.5 occurs when the total score \( T = 5 \) from pairs like \( (1,4) \), \( (2,3) \).

Understanding average scores is beneficial for interpreting general trends and expectations from the collected data without focusing on individual details.