A total of 4 buses carrying 148 students from the same...

A total of 4 buses carrying 148 students from the same school arrives at a football stadium. The buses carry, respectively, $40,33,25$, and 50 students. One of the students is randomly selected. Let $X$ denote the number of students that were on the bus carrying this randomly selected student. One of the 4 bus drivers is also randomly selected. Let $Y$ denote the number of students on her bus. (a) Which of $E[X]$ or $E[Y]$ do you think is larger? Why? (b) Comnute $F[X]$ and $E[Y)$

A total of 4 buses carrying 148 students from the same school arrives at a football stadium. The buses carry, respectively, $40,33,25$, and 50 students. One of the students is randomly selected. Let $X$ denote the number of students that were on the bus carrying this randomly selected student. One of the 4 bus drivers is also randomly selected. Let $Y$ denote the number of students on her bus.
(a) Which of $E[X]$ or $E[Y]$ do you think is larger? Why?
(b) Comnute $F[X]$ and $E[Y)$

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Key Concepts

Expected Value

The expected value is the long?run average of a random variable, computed as the sum of all possible outcomes weighted by their probabilities. In problems like this, calculating the expected value requires careful assignment of probabilities to each outcome, whether the sampling is done uniformly or in a size?biased manner.

Size-Biased Sampling

Size-biased sampling occurs when the probability of selecting a group (or an item within a group) is proportional to the group's size. For example, when a student is chosen at random, larger groups (buses with more students) are more likely to be represented. This contrasts with uniform sampling, where each group is selected with equal probability regardless of its size.

Probability Mass Function

The probability mass function (pmf) for a discrete random variable assigns a probability to each possible value the variable can take. In the context of this problem, the pmf for the random variable representing the bus size of a randomly selected student is determined by the proportion of students in each bus relative to the total, while the pmf for the bus size of a randomly selected bus driver assigns equal probability to each bus.

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