Answer :
Final Answer:
The correct expression for the quantile, Qp, where p ∈ [0, 1], is Qp = p(1 - p). Option c is the answer.
Explanation:
A quantile, denoted by Qp, divides a probability distribution into equal-sized portions. The value p represents the proportion of the distribution that falls below Qp.
Here's why option (c) is the correct formula:
p: This represents the portion of the distribution less than the quantile (Qp).
(1 - p): This represents the remaining portion of the distribution greater than the quantile (Qp).
Multiplying these two terms, p(1 - p), gives the probability that a value falls within the specific quantile range. This aligns with the concept of a quantile dividing the distribution into proportional areas.
The other options are incorrect because:
a) Qp = 2p: This would not limit the value between 0 and 1 for all quantiles (0 ≤ p ≤ 1).
b) Qp = p²: This wouldn't provide equal-sized portions for the distribution as the values would be skewed towards the lower end (0 ≤ p ≤ 1).
d) Qp = p^2: Similar to option (b), this formula would also lead to skewed quantile values.
Option c is the answer.
