I'm trying to help a friend with a uni exam he failed in December but I can't seem to work out this next question (translating from french, sorry for any mistakes):
Consider a first order continuous-time dynamic system:
$$ \dot x = x \ , \quad x \in \mathbb R $$
where the condition $x(0)$ at the instant $t=0$ is distributed following the probability density in the image below:
Let $U=[0.15, \ 0.6, \ 0.53, \ 0.9, \ 0.34, \ 0.17, \ 0.82, \ 0.08, \ 0.37, \ 0.03]$ a sequence of 10 random numbers pulled from a uniform distribution between 0 and 1.
Estimate using the sequence $U$ and Monte Carlo:
The average of the solution $x(1)$
The probability that $x(1)>0.3$
Can anyone give me any tips or point me to some relevant resources online?
Thanks in advance :)