Derive distribution of an inverse transform

Q: Derive distribution of an inverse transform

This question evaluates change-of-variables and inverse-transform techniques in probability, focusing on the logit (inverse logistic) mapping, Jacobian-based density transformation, handling of support, and computation of distributional moments.

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Question

Change of Variables via the Logistic Map

You are given a random variable X with density f_X supported on (0, 1). Define the strictly increasing logistic map

g(u) = 1 / (1 + e^{−u}),

which maps the real line ℝ onto (0, 1). Let U = g^{−1}(X) be the logit transform of X.

Tasks

(A) Derive a general formula for the density of U = g^{−1}(X) in terms of f_X and g. Clearly state the support.

(B) Specialize to X ~ Uniform(0, 1). Compute the distribution (CDF and PDF), mean, and variance of U = log(X / (1 − X)). Show the Jacobian steps and the support.

Derive distribution of an inverse transform

Change of Variables via the Logistic Map

Tasks

Solution

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Derive distribution of an inverse transform

Overview

Change of Variables via the Logistic Map

Tasks

Solution

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