High School

Find an expression for the quantile, \( Q_p \), where \( p \in [0,1] \).

A. \( Q_p = 2p \)

B. \( Q_p = p^2 \)

C. \( Q_p = p(1-p) \)

D. \( Q_p = p^2 \)

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.