Final answer:
The random variable Y follows a beta distribution, indicated by its density function f(y) = cy³(1−y)⁸₀ with the parameters Ø=4 and β=81, commonly used to model proportions such as the time checkout counters are busy.
Step-by-step explanation:
The random variable Y described in the question seems to follow a beta distribution, indicated by its density function f(y) = cy³(1−y)⁸₀. In a beta distribution, the probability density function is given by f(y) = γyʸ−Ø(1 - y)ʹ−β for y between 0 and 1, where γ, Ø, and β are parameters that shape the distribution. Here, the exponent of y is 3 (i.e., Ø = 4), and the exponent of (1 - y) is 80 (i.e., β = 81), suggesting a beta distribution with parameters Ø=4 and β=81 after properly normalizing the constant c to make sure that the total area under the curve is 1.
The beta distribution is commonly used in scenarios where we model the behavior of random variables that represent proportions, such as the proportion of time that checkout counters are busy. It is bound between 0 and 1 and is particularly flexible, allowing for a variety of shapes depending on the parameters chosen. Understanding the shape of the distribution is important for tasks such as estimating probabilities, calculating expectations, and conducting hypothesis testing about proportions.