Let's assume $\Omega=[0,1]$ and $F$ is a Borel set of $[0,1]$, and $\mathbb{P}$ is Lebesgue's measure in $[0,1]$. Also let's assume that $\forall n \geq 1$, $X_n$ is a random variable defined such $X_n(\omega)=\omega^n$ and of course $\omega \in \Omega$.
How do I define the density of $X_n$ ?
I've tried to work on its repartition function like this :
$\mathbb{P}(X_n \leq t)= \int_{[0,1]} ω^n.\mathbb{1}_{\omega \leq t} d\omega$
then discussing both cases when $t \in [0,1]$ or not, but I seem to find zero which is weird since it should be something that depends on t that I would after that derive to finally find my density function.
What seems to be wrong in my work ? can you help me find my error please ?
The probability that $$X_n(\omega)=\omega^n\leq x$$ is the Lebesgue measure of the set
$$\{\omega \in [0,1] | \omega^n\leq x \}=\{\omega\in [0,1] |\omega\leq x^{\frac 1 n}\}$$ which is
$$x^{\frac 1 n}, x\in [0,1],$$
and zero below $0$, and $1$ above $1$.
The derivative of the function above is the density of $X_n$.
Your mistakes are shown in red below:
$$\mathbb{P}(X_n \leq t)= \int_{[0,1]} \color{red}{ω^n}.\mathbb{1}_{\omega^{\color{red}n} \leq t} d\omega$$