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Variance tells us about the spread or dispersion of a set of data points. Think of it like the average of the squared differences from the mean. In probability theory, having a high variance indicates that the data points are spread out over a larger range. For the professor's exam setup where each question is answered correctly with probability \( p_j \), the variance of the total score \( S_n \) helps us understand how much the number of correct answers is expected to deviate from the mean, which is \( 0.7n \). The variance is calculated as \( Var(S_n) = \sum_{j=1}^n p_j(1-p_j) \). This indicates how much the scores will vary if the probabilities are different for each question.
For a true-false exam with all questions having \( p_j = 0.7 \), the variance becomes \( 0.21n \), showing that there's some level of certainty in how many questions the students will answer correctly, but it's not as uniform as it could be with other distributions.

Expected value is a fundamental concept in probability, referring to the average value you would expect from repeated experiments. Consider it like the center of gravity in probability terms. For the exam, each question has an associated probability \( p_j \) that it will be answered correctly. The expected total number of correct answers \( E(S_n) \) is the sum of these individual probabilities.
We calculate \( E(S_n) = \sum_{j=1}^n p_j = 0.7n \), meaning on average, students will get \( 70\% \) of the questions right. The professor aims for this expectation to be consistent across all students, regardless of the individual probability assigned to each question. By maintaining \( p_j = 0.7 \) for every question, she aligns each question's difficulty level with the target expected outcome.

In this scenario, each true-false question operates as an independent experiment. This means the outcome of one question doesn't affect the outcomes of others. Independence is a key aspect of probability that simplifies calculations, as it allows us to treat each question's probability separately.
If the questions were somehow dependent on each other, the probabilities involved would be trickier to estimate, altering our calculations for expected value and variance. By treating the questions independently, the professor can confidently assert that the total expected correct answers and their variance can be derived by summing the potential outcomes of each question without considering interactions between questions.

A Bernoulli variable is a type of random variable with two possible outcomes: success (usually coded as 1) or failure (coded as 0). In the context of a true-false test, each question answered correctly can be viewed as a Bernoulli trial with probability \( p_j \) being the chance of success.
For a Bernoulli variable \( X_j \), its expected value is \( E(X_j) = p_j \) and its variance is \( Var(X_j) = p_j(1-p_j) \). When we talk about the student's score \( S_n \), it is essentially the sum of these Bernoulli variables across \( n \) questions.
Thus, the professor's challenge is to choose \( p_j \) values that satisfy both the desired expected value of the score and a sufficiently large variance. Settling with \( p_j = 0.7 \) for all questions ensures the mean score aligns with the target, while also keeping the variance reasonably high.