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The expected value, often referred to as the mean of a probability distribution, is a key concept in probability and statistics. It provides an average outcome if an experiment could be repeated infinitely. For a continuous random variable like our example with the proportion of time all checkout counters are busy, we calculate the expected value \[ E(Y) = \int_0^1 y \cdot f(y) \, dy \]where \( f(y) \) is the probability density function (PDF) of \( Y \). In this particular example, after determining the constant \( c \) such that the PDF is valid, the expected value is found to be \[ E(Y) = \frac{35}{9}. \]This value represents the weighted average, where \( y \) values are weighted by their probability \( f(y) \). In context, this means that on average, the checkout counters are busy a certain proportion of the time each day.

The Beta function is a special mathematical function that is very useful in probability and statistics. It is denoted as \( B(x, y) \), and it helps evaluate integrals that arise from continuous probability density functions when they resemble expressions like \[ B(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1} dt. \]This function is particularly effective when solving integrals related to expressions of the form \[ y^{a}(1-y)^{b} \]as it matches the Beta function's structure. In our solution, the Beta function is used to find constants such as \( c \) and to compute expected values. The neat property of the Beta function is its connection to the Gamma function, another important function which simplifies calculations tremendously when used together, further explained below.

The Gamma function, denoted \( \Gamma(n) \), is a fascinating mathematical tool that extends the concept of factorial to non-integer numbers. For positive integers, it's defined as \[ \Gamma(n) = (n-1)! \]but it also supports fractional and real numbers. The connection between the Gamma and the Beta function is given by the relation\[ B(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}. \]In our example, we used the Gamma function to solve the integral by first expressing it in terms of the Beta function and then evaluating the Beta function using the Gamma function values. This process highlights the deep relationship between these functions, enabling the evaluation of complex integrals effortlessly and obtaining solutions to probability-related problems efficiently.