How can I determine the distribution of variables?

Question

Suppose that $X_i \sim N(\mu,\sigma^2)$ with $i = 1, … , n$ and $Z_i \sim N(0,1)$ with $i = 1, … ,k$, and all variables are independent. I have to find the distributions of the following variables:

1. $\frac{X_1 - X_2}{\sigma S_z \sqrt2}$
2. $\frac{\sum_{i=1}^{n} (X_i - \mu)^2}{\sigma^2} + \sum_{i=1}^{k} (Z_i - \bar{Z})^2$
3. $k\bar{Z}^2$

I understand that we must use standardizing, the chi-squared distribution, F-distribution and the student t distribution in order to solve these. However, I can't fully grasp the way of approaching these kind of questions and to manipulate the variables such that a distribution arises. Could someone point me in the right direction? If something is not clear, tell me and I will gladly further clarify.

Answer 1

You have to manage your expressions in order to find the definition of known distributions as

• Student t

• $\chi_{(n)}^2$

• Fisher F

and so on...

1. let's start with the first example:

Observe that

$$\frac{X_1-X_2}{\sigma\sqrt{2}}\sim Z$$

thus your ratio 1. is

$$\frac{Z}{\sqrt{\frac{Y}{(k-1)}}}$$

Where $Y=\frac{(k-1)S_{Z}^2}{1} \sim \chi_{(k-1)}^2$ and $Z\perp\!\!\!\perp Y$

The independence between $Z,Y$ is stated by hypothesis but anyway, in a Gaussian model, this is granted by Basu's Theorem and Cochran's Theorem

This is exactly the definition of a Student's t distribution with $(k-1)$ d.o.f.

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2. Just observe that the two addend are both chi-squared distributions where

$$\frac{\sum_{i=1}^n(X_i-\mu)^2}{\sigma^2}\sim \chi_{(n)}^2$$

Here the mean is a known value....

and

$$\sum_{i=1}^{k}(Z_i-\overline{Z})^2\sim \chi_{(k-1)}^2$$

Thus their sum, using additivity of Gamma distributions is a $\chi_{(n+k-1)}^2$

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3. observe that

$$\overline{Z}\sim N\left(0;\frac{1}{k}\right)$$

which means that

$$\sqrt{k}\cdot\overline{Z}\sim N(0;1)$$

and thus

$$k\overline{Z}^2=\left(\sqrt{k}\cdot\overline{Z}\right)^2\sim \chi_{(1)}^2$$