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In my documentation they explain me how to calculate specific probabilities of some statistics

like $P(\bar X \ge x)$ or $P(S^2\ge x) $ knowing $X$~$N(\mu,\sigma)$

but then in the exercices they keep asking me things like this:

$P(S^2/\sigma^2 \le 2.041) $ with $ n=16$ and $X$~$(\mu,\sigma)$

or

$P(-1 \le [\bar X- \bar Y-(\mu_X-\mu_Y)]\le1)=95$% knowing: $\sigma^2_X=2$ and $\sigma^2_Y=2.5$ and $k=m=n$ (both are $N(\mu,\sigma)$ (find k)

How can I know the probability of a combination of statistics?( like those two examples) Is there any general rule to calculate a a combination of statistics for each distribution?

Thanks.

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    In general you can use conditioning type arguments to solve these types of problems. In the case of normal distributions you just need to use the fact that linear combinations of normal random variables are normally distributed: onlinecourses.science.psu.edu – 2017-01-16
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    Thanks but how would yo solve the first one then? I mean wich formula or wich table (like chi-squared) because if i try to bring it to a normal standard distribution to look at the table I lack the $\sigma& to find the value – 2017-01-16
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    You can use the fact that $\frac{(n-1)S^2}{\sigma^2} \sim \chi^2 (n-1)$ – 2017-01-16
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    Oh, silly me, i didn't realise I could just (16-1)*2.041. Thank you very much now im starting to get it better. – 2017-01-16

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