Rules of algebra with Normal distribution

Question

As part of my self study for Financial Economics, I came across simplifying the following expression,

$$\sigma Z_{0.25} + (2\sigma)(Z_{0.5} - Z_{0.25})$$

where $Z$ are independent Brownian motion ($Z_0 = 0$ and $Z_t-Z_s \mathtt{\sim} N(0, t-s)$.

It is simplified as,

$$\sigma N(0, .25) + 2\sigma N(0, .25) = N(0, .25\sigma^2) + N(0, .25 \times 4\sigma^2)$$ $$= N(0, 1.25\sigma^2) = \sqrt{2.5}\sigma Z_{0.5}$$

However, if I simplify the first expression to $\sigma (2Z_{0.5} - Z_{0.25})$, I get a different Normal distribution, $N(0, 1.75\sigma^2)$ which is $\sqrt{3.5}\sigma Z_{0.5}$.

Edit: Here's how I did the above simplification,

$$\sigma Z_{0.25} + (2\sigma)(Z_{0.5} - Z_{0.25})$$ $$= \sigma Z_{0.25} + 2\sigma Z_{0.5} - 2\sigma Z_{0.25}$$ $$= 2\sigma Z_{0.5} - \sigma Z_{0.25}$$ $$= \sigma (2 Z_{0.5} - Z_{0.25})$$

using basic algebra.

So, obviously, the Brownian motion variables and Normal distributions don't support all of algebra. So, what subset of the algebraic rules are allowed when dealing with these? Is there somewhere these are set out that I can refer to? Thank you.

Answer 1

$2\sigma (Z_{0.5}-Z_{0.25})\neq \sigma (2Z_{0.5}-Z_{0.25})$

$2\sigma (Z_{0.5}-Z_{0.25})= \sigma (2Z_{0.5}-2Z_{0.25})$

You just forgot to distribute the 2 to both terms in the parentheses. The textbook is definitely correct and you definitely don't need any special math to simplify that expression.

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edit: I see your problem. You're actually right in the sense that you can't just subtract off coefficients on random variables. Recall what "subtracting off" actually means in terms of 6th grade algebra. $2x-x=x$ if and only if $2x = x + x$. In this case, $2\sigma Z_{0.25} \neq \sigma Z_{0.25} + \sigma Z_{0.25}$.

To convince yourself, try simplifying the distributions (i.e. plugging in the variances) in $\sigma Z_{0.25} + 2\sigma Z_{0.5} - 2\sigma Z_{0.25}$. You get $Z_{.25*\sigma^2} + Z_{2\sigma^2} - Z_{\sigma^2}=Z_{.25*\sigma^2} + Z_{2\sigma^2} - Z_{\sigma^2}$.

The "rules of algebra" still apply where they're relevant, but in this case it takes some care not to apply them where they aren't. Then again I probably would done the same thing in your position.