Hallo,
$\displaystyle X_1, X_2, X_3$ are iid random variables with a lognormal distribution.
Am I right, that
$\displaystyle f(X_1),f(X_2),f(X_3)$ where $\displaystyle f(x)=(1+x)^\gamma, 0< \gamma < 1$
are also independent?
Thanks in advance!
Hallo,
$\displaystyle X_1, X_2, X_3$ are iid random variables with a lognormal distribution.
Am I right, that
$\displaystyle f(X_1),f(X_2),f(X_3)$ where $\displaystyle f(x)=(1+x)^\gamma, 0< \gamma < 1$
are also independent?
Thanks in advance!
Hello,
Yes, just use the definition of independence : X and Y are independent iff for any *correct* functions f,g, E[f(X)g(Y)]=E[f(X)]E[g(Y)]
*correct* in the sense that the expectations are defined for such functions. That would be measurable and bounded for the simplest situation I guess...
You can also prove that
$\displaystyle P[f_1(X_1) \in B_1, ..., f_n (X_n) \in B_n] = P[f_1 (X_1) \in B_1] \cdots P[f_n (X_n) \in B_n]$
where the $\displaystyle B_i$ are arbitrary 1d Borel sets. This implies independence from a different starting place; all my references use a different definition of independence than Moo's, but they are equivalent.
Thank you. So if I show it this way, I just need the inverse function and use the independence of$\displaystyle X_1,X_2,X_3$ and then it follows.
Am I right?
Doesn't it matter that my function $\displaystyle f(x)=(1+x)^\gamma,0<\gamma<1, $ is just defined for $\displaystyle x\geq -1$ and the inverse function for $\displaystyle x \geq 0$?