Four buses carrying 148 students from the same school arrive...

Four buses carrying 148 students from the same school arrive 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 the 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) Compute $E[X]$ and $E[Y]$

Four buses carrying 148 students from the same school arrive 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 the 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) Compute $E[X]$ and $E[Y]$

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Key Concepts

Uniform Sampling

Uniform sampling is the process of selecting an outcome from a set where every element has an equal probability of being chosen. When a bus driver is selected uniformly from a set of drivers, each driver (and hence each bus) is equally likely to be chosen, regardless of the size of the bus.

Size Bias

Size bias occurs when the method of selecting a sample makes larger groups more likely to be chosen than smaller ones. This is evident in the scenario where a student is picked at random, which results in a higher probability of drawing a student from a larger bus, hence inflating the expected number of students compared to a uniform selection.

Expected Value

Expected value is a fundamental concept in probability that represents the average outcome of a random variable over numerous repetitions of an experiment. It is calculated as the weighted sum of all possible values of the variable, with each value multiplied by its probability of occurrence.

Weighted Sampling

Weighted sampling refers to selecting an outcome where the probability of selection is not equal for all elements, but rather depends on a weighting factor. In many cases, such as sampling a student from a bus, the likelihood of choosing a particular bus is proportional to the number of students on that bus, leading to a size-biased effect.

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