I am modelling some prices and I am using a t-distribution. I would like to have the opportunity to change my mean parameters but keeping the same variance/volatility. So for example originally my mean price is 35,000 USD, but I would like to have the same distribution only with the mean 70,000 USD.
How do I transform the second parameter (sample standard deviation)?
In Gaussian, since $Var(2X) = 4Var(X)$, $std$ would be just $\sqrt{4*var (X)}=2\sigma$, right?
Does it apply to the t-distribution as well?
Thanks
To answer your question simply, Yes. Multiply S by 2 times. Please see if the below explanation helps:
Long Answer
Let $X_1,\dots X_n$ be independently and identically drawn from the distribution $N(\mu, \sigma^2)$
$\bar X = \frac 1 n \sum_{i=1}^n X_i$ be the sample mean and let $S^2 = \frac 1 {n-1} \sum_{i=1}^n (X_i - \bar X)^2$ be the sample variance.
Then the random variable $\frac{ \bar X - \mu } { S /\sqrt{n}}$ has a Student's t-distribution with n - 1 degrees of freedom.
Now let's $X_1,\dots X_n$ be $\sim c$ times the earlier values, for some constant $c\in \mathbb{R}_+$. Then $\bar X$ will be $\sim c$ times the earlier values. Now if $S^2$ will be $\sim c^2$ times the earlier values, then the random variable $\frac{ \bar X - \mu } { S /\sqrt{n}}$ still has Student's t-distribution with n - 1 degrees of freedom, as the constant terms $c$ on both the numerator and denominator cancel out. Thus S is $c$ times it's original value