# Determining Value at Risk of a Poisson distribution

If my discrete random variable had a poisson distribution with both moments say equal to 10, how can I find the Value at Risk for a 95 percent confidence interval?

I have seen that I need to integrate the PDF from the lower limit up until $$L$$ so I am trying to integrate that from $$0$$ to $$L$$ then equating it to 0.5 but its an absolute mess. Any help would be appreciated.

Firstly, a Poisson distribution is discrete one, so you can get CDF by sumation instead of integration. See here how CDF looks like.

Secondly, 95 % VaR is a opposite value to 5 % quantile of the distribution (i.e. if quantile is -10, VaR is +10, meaning loss 10), so have a look at some statistical table. For example here.

• okay, that makes a lot more sense because its discrete, how do i go about reading this table for this problem? would this correspond to x=5 with mean value =10?
– user46739
May 7 '20 at 7:56
• @123123: Mean, i.e. $\lambda = 10$ is on second page, then go down in the table until you reach 0.05. This is between $x=4$ and $x=5$. So, VaR is between these two values, closer to $x=5$ because 0.05 is closer to 0.0671 than 0.0293. May 7 '20 at 8:47
• amazing thank you so much, however this would set our Var to be -5, i thought that Var was strictly positive?
– user46739
May 7 '20 at 9:20
• Random variable with Poisson distribution can be only positive or zero (see link to Wiki where the support is set $\mathbb{N}_0$). Generally VaR is defined as opposite value to the value of the quantille because VaR is a loss. So if the quantile is negative, VaR is positive meaning the loss. In case the quantile is positive, VaR is negative, meaning we have negative loss, i.e. gain. In your case the interpretation is that with probability 95 %, your gains will be higher than 5. May 7 '20 at 10:57
• Just note, the VaR is exactly -5. My previous claim that it is between 4 and 5 is fault because the distribution is discrete. May 7 '20 at 11:04