Question

Consider relations between three well-known distributions:

(a) Let U be uniform on (0,1), and define X = -log(U). Find the possible values of X and the cumulative distribution function (cdf) of X. Recognize that X has a well-known distribution, provide its name, and specify its parameters.

(b) Products of uniform (0,1) random samples can be represented as the geometric mean of the sample. Let U1, U2, ..., Un be an i.i.d. uniform (0,1) sample, and define Yn = (U1 * U2 * ... * Un)^(1/n). Show that when n is large, the distribution of log(Yn) is close to a famous distribution. Identify its name and parameters. [Utilize the result from Part a.]

(c) Let X be normal (μ, σ^2), and define W = e^X. The distribution of W is called lognormal with parameters μ and σ^2. Find the cumulative distribution function of W in terms of the standard normal cdf Φ, covering the entire real line.

(d) In the context of Part b, determine the approximate distribution of Yn when n is large. Consequently, find the approximate cumulative distribution function of Yn for large n, without using integrals, but you can use the standard normal cdf Φ.

Answer 1

The distribution of X defined as -log(U) for U uniform on (0,1) follows an exponential distribution. For large n, log(Yn), where Yn is the geometric mean of an i.i.d. uniform (0,1) sample, resembles a normal distribution. W = e^X, when X is normally distributed, is lognormally distributed, and its CDF can be expressed using the standard normal CDF Φ.

Step-by-step explanation:

Understanding Probability Distributions

Let's explore these concepts step by step:

a. If U is uniform on (0,1), then by defining X = -log(U), X can take all positive values. The cumulative distribution function (CDF) of X is given by P(X ≤ x) = P(-log(U) ≤ x) = P(U ≥ e^{-x}) = 1 - P(U < e^{-x}) = 1 - (e^{-x} - 0)/(1 - 0) = 1 - e^{-x}, for x ≥ 0. X follows an exponential distribution with parameter λ = 1.

• b. For the product of an i.i.d. uniform (0,1) sample, as n becomes large, the Central Limit Theorem implies that the distribution of log(Yn) converges to a normal distribution, due to the additive property of logarithms of the product. Utilizing the result from Part a, we find the distribution resembles the normal distribution with the mean and variance obtained from the exponential distribution of -log(U).
• c. For X being normal with mean μ and variance σ^2, and W = e^X, the CDF of W is given by P(W ≤ w) = P(e^X ≤ w) = P(X ≤ log(w)) = Φ((log(w) - μ) / σ). This covers all values of W since X can span the entire real line.
• d. Approximating the distribution of Yn for large n, it is lognormal based on the behavior of log(Yn) and its relation to the normal distribution. The approximate CDF of Yn without using integrals is thus P(Yn ≤ y) = Φ((log(y) - μ) / σ), where μ and σ are the mean and standard deviation derived from the underlying normal distribution of log(Yn).