Question

Let 0 < p < 1. Find the 0.20 quantile (20th percentile) of the distribution that has pdf f(x) = 4x^3, 0 < x < 1, zero elsewhere.

Answer 1

To find the 20th percentile of the given distribution, one must integrate the pdf to get the CDF, set that equal to 0.20, and solve for x, which yields the fourth root of 0.20 as the answer.

Step-by-step explanation:

To find the 0.20 quantile (20th percentile) of the distribution with the probability density function (pdf) given by f(x) = 4x3, 0 < x < 1, we need to solve for x such that the cumulative distribution function (CDF) equals 0.20.

The CDF is found by integrating the pdf from 0 to x:

• Find the indefinite integral of the pdf: F(x) = ∫ 4x3 dx = x4 + C.
• Since F(0) = 0, we determine that C = 0, so F(x) = x4.
• Solve for x when F(x) = 0.20: 0.20 = x4.
• Calculate the fourth root of 0.20: x = 0.201/4;
• The 0.20 quantile is x = √√√√0.20

Answer 2

Calculate the CDF by integrating the PDF from 0 to x, then solve for x when F(x) equals 0.20 to find the 0.20 quantile (20th percentile) of the distribution.

Step-by-step explanation:

To find the 0.20 quantile of the distribution with the given probability density function f(x) = 4x^3, where 0 < x < 1, we first calculate the cumulative distribution function (CDF) by integrating the PDF.

The CDF F(x) will be the integral of 4x^3 from 0 to x, which gives us F(x) = x^4 for 0 < x < 1. To find the 20th percentile,

we solve F(x) = 0.20, resulting in x = √[0.20], which simplifies to x = 0.20^(1/4).

After computing this, we obtain the quantile as x ≈ 0.6687, which is the value at which the area under the probability curve from 0 up to this value comprises 20% of the total area.