t Distribution – Advanced

Property A: If z has distribution N(0, 1), u2 has distribution χ2(m) and z and u are independent, then

image3357

has distribution T(m).

Proof: Since z has distribution N(0, 1), any linear combination of z is also normal, and in particular y = z\sqrt{m} has distribution N(0,\sqrt{m}). Let f the pdf for y. Therefore

image3361

Let x = u2. Thus u\sqrt{x}, and so using the change of variables technique (Theorem 2 of General Properties of Distributions), if the pdf of x is h, then the pdf g of u is

image3364

But since x = u2 has distribution χ2(m), we have

image3366

Since t = y/u is an increasing function of y, keeping u fixed, if k(u, y) is the joint frequency function of u and y, and h(u, t) is the joint frequency function of u and t, then by a corollary to the change of variables technique,

image3370

Since y and u are independently distributed and y = tu,

image3372

Thus,image3373

It now follows that the pdf q(t) of t is given by

image3375

where
image3376

Let
image3377

Then
image3378

and so
image3379

Thus,
image3380image3381 image3382 image3383

which completes the proof.

Theorem 1: If x has normal distribution N(μ, σ), then for samples of size n, the random variable

image674

has distribution T(n – 1).

Proof: Since x has normal distribution, by the Central Limit Theorem the sample mean has normal distribution \sigma/\!\sqrt{n}, and so z has distribution N(0, 1) where

image428

By Corollary 3 of Chi-square Distribution, for samples of size n the sample variance s2 has distribution

image3385

Now define the random variable  as follows:

image3386

It now follows that u has distribution χ2(n–1). Finally define the random variable t as follows

image3388

From Property A, since and s2 are independently distributed, it follows that t has distribution T(n–1). But also,

image3390

2 Responses to t Distribution – Advanced

  1. Cristian Gradinaru says:

    Charles, I believe you have an error above, “Let x = u^2. Thus u = z * sqrt{x}, and so using the change of variables…”

    If x=u^2 then that means u = sqrt(x), not z * sqrt(x).

    regards,
    Cristian

    • Charles says:

      Cristian,
      Thanks for catching this typo. I have just corrected the webpage per your suggestion. I appreciate your help in making the website better.
      Charles

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