Suppose $X$ is a $\chi_k^2$-distributed random variable, then is there any explicit form for the probability $$\mathbb{P} (X < k)?$$ In particular, I'm interested in the asymptotic value of $$\lim_{k\to \infty}\mathbb{P} (X < k). $$ Thank you.
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Using the central limit theorem, you can show that this asymptotic value is $0.5$. – Augustin Aug 30 '15 at 15:36
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According to the central limit theorem $\lim_{k \to \infty}\chi_k^2\sim\mathcal N(k,2k)$ – callculus42 Aug 30 '15 at 15:41
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The asymptotic value is $\frac{1}{2}$.
Let $X_i$ be an i.i.d. sequence of standard Gaussian random variables. Then $Y_k = \sum \limits_{i = 1}^k X_i^2 \sim \chi^2_k$. Now note that
$$P(Y_k < k) = P(Y_k \le k) = P\left(k^{-1/2}(Y_k - k) \le 0\right) \xrightarrow{k \to \infty} \frac{1}{2}$$ because $k^{-1/2}(Y_k - k) \xrightarrow{d} \mathcal{N}(0, 2)$ by the central limit theorem.
Dominik
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