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Assume $X_n$ is distributed $\text{Beta}(1/n, 1/n)$ and $X$ is distributed as $\text{Binom}(1,1/2)$. Show that $X_n$ converges to $X$ in distribution.

I'm having some issue with this question. I understand that if I want to show that if $X_n$ converges to $X$ in distribution then $F(X_n) \rightarrow F(x)$. This is for their cumulative distribution functions. However, I'm a bit stuck on how exactly to go about this.

whuber
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Warhawk1987
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    Hi, welcome. Perhaps also add the `self-study` tag. – Jim Sep 13 '19 at 18:30
  • Your understanding is almost correct if we re-interpret your notation: you need $F_{X_n}(x)$ to converge to $F_X(x)$ for every $x$ where $F_X$ is continuous. Convergence usually will not occur wherever $F_X$ has a jump. That simplifies your task, because it means you only need to consider $x$ in the interval $(0,1).$ – whuber Sep 13 '19 at 19:03

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