Question

If u is uniform on [0, 1], find the density function of √ u.

A) 2√u
B) 1/√u
C) √u
D) 1/(2√u)

Answer 1

The density function of √u when u is uniformly distributed on [0, 1] is 2√u.

Step-by-step explanation:

To find the density function of √u when u is uniformly distributed on the interval [0, 1], we can use the method of transformation.

Let's consider the cumulative distribution function (CDF) of √u, denoted as F(√u). The CDF of u is F(u) = u for u in [0, 1]. Therefore, the CDF of √u can be calculated as:

F(√u) = P(√u ≤ x) = P(u ≤ x^2) = F(x^2) = x^2,

where x is a value in the interval [0, 1]. Finally, the density function of √u is the derivative of F(√u) with respect to x:

f(√u) = d/dx(x^2) = 2x.

Therefore, the density function of √u is 2√u.