# Sample from sum of squared independent $t_\nu$ distributed variables

I would like to specify radius r in a case where: $$\mathbb{P}(Z_1^2 + Z_2^2>r)=a\,,$$ $a$ is known, and $Z_1,Z_2$ are i.i.d $t_\nu$-distributed.

Question: Any ideas on how to do this?

Attempt: Since this can be really challenging to do analytically, I decided to do it by sampling.

My first thought was to simply draw N random values a from $t_\nu$ distributed variable and then count how many are there that fulfill the $|t_k|\geq r$ condition, then update $r$ until I cross $a$, but I'm not really convinced that this is the correct way. The $\nu$ value doesn't really matter.

I have checked these two posts: (1) (2).

Edit As was pointed out in the comments by whuber, it may be possible to numerically compute the probability through the integral.

I would also use this solution, but as I understood from the posts linked above, it will be really tricky.

• Your question--which asks to compute the inverse distribution function of a sum of squared t variables--does not seem to agree with the title, which asks to draw a sample. Could you please edit one (or both) to make them consistent? – whuber Dec 2 '15 at 17:46
• If that calculation would help you, then please edit your question to mention that. I'm confident that many community members would know of answers and I believe you're likely to get several different, but good, ones. – whuber Dec 2 '15 at 18:12
• This PDF is benign! There isn't anything terribly tricky about it. There are many ways to perform the computation accurately and fairly quickly. – whuber Dec 2 '15 at 18:49
• Sure: for $\nu=3$ and $r=3.5$, for example, the complementary integral can be found by Wolfram Alpha – whuber Dec 2 '15 at 20:43
• The event $\{Z_1^2+Z_2^2\ge r\}$ is also an event for $(Z_1,Z_2)$ so all you need is a pdf on $(Z_1,Z_2)$. – Xi'an Dec 3 '15 at 13:32