Question

Let X₁, X₂,....Xₙ, be a random sample from a distribution that can take on only positive values. Use the Central Limit Theorem to produce an argument that if n is sufficiently large, then Y=X₁,X₂,...Xₙ, has approximately a lognormal distribution.

Define Wᵢ, = In Xᵢ for i = 1,2,....n By the Central Limit Theorem, Z = ....
(In X₁+ In X₂+ .... + In Xₙ) = .......
This means ... = In Y, so Y has a lognormal distribution.

Answer 1

The Central Limit Theorem indicates that the sum of natural logarithms of positive random variables is approximately normally distributed as sample size increases. Since the log of the product of these variables (Y) is normally distributed, Y is lognormally distributed.

Step-by-step explanation:

The question is asking how to use the Central Limit Theorem to argue that if n is large enough, the product of random variables X₁, X₂,...,Xₙ will have an approximately lognormal distribution.

The Central Limit Theorem states that the sum of a large number of independent random variables, each with finite mean and variance, will be approximately normally distributed.

Let's define Wᵢ as the natural logarithm of each positive random variable Xᵢ. Given that Xᵢ can only take positive values, Wᵢ will always be defined. If we sum the Wᵢ variables for i = 1 to n, the sum will approach a normal distribution as n becomes large. This can be expressed as: Z = (ln X₁ + ln X₂ + ... + ln Xₙ).

Since the natural logarithm is a continuous and one-to-one function, if we exponentiate both sides of the equation, we get e to the power of Z, which is the original product Y = X₁ X₂ ... Xₙ. As the log of Y is normally distributed, Y itself is lognormally distributed.