I have a little issue : I want to prove that if $\{X_n\}_{n \in \mathbb N}$ is a sequence of random variables defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that $\forall n \in \mathbb{N}, X_n \sim \mathcal{T}_n$ then $\forall \alpha > 0, \{\mathbb{P}(X_n \leq \alpha)\}_{n \in \mathbb{N}}$ is an increasing sequence.

I tried to express this probability using the density function : $$\forall \alpha > 0, \mathbb{P}(X_n \leq \alpha) = \frac{1}{2} + \int_{0}^\alpha f_n(t) d\lambda(t)$$ when $f_n$ stands for the density (according to Lebesgue measure) of $X_n$ and then use results on integrale depending of a parameter but it led to a lot of calculations...


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