# How to find the Percentile of this distribution?

Let say, we have 2 IIDs $$X, Y \sim N \left( 0, 1 \right) + \eta$$

Now $$\eta$$ has a discrete distribution with values 0(-10) with probabilities 0.991 & 0.009 respectively.

Also assume that $$Z = X+Y$$.

I need to find the value of $$z$$ such that $$P \left[ Z < z \right] = 1\%$$

Is there any closed form solution available for the value of $$z$$?

• There is no closed form solution, but the value is relatively easy to find numerically by finding the (unique zero of the) function $\Pr(Z \lt z) - 1/100.$
– whuber
Nov 2, 2020 at 23:06

Another way to express these quantities is to let $$W$$ be a standard Normal variable and $$U$$ be a Bernoulli$$(0.009)$$ variable. Both $$X$$ and $$Y$$ have the distribution of $$W - 10U.$$ Thus, $$Z=X+Y$$ has the distribution of (a) the sum of two iid standard Normal variables plus $$-10$$ times (b) the sum of two iid Bernoulli$$(0.009)$$ variables.
It is elementary that (a) has a Normal$$(0,\sqrt{2})$$ distribution and (b) has a Binomial$$(0.009, 2)$$ distribution. This latter takes on three values $$0,1,2$$ with chances $$(1-p)^2, 2p(1-p),$$ and $$p^2,$$ respectively (writing $$p=0.009$$). Subtracting $$10$$ times their value exhibits $$Z$$ as a mixture of three Normal variables with means $$0$$, $$0-10(1)=-10,$$ and $$0-20(1)=-20.$$ The mixture weights are $$(1-p)^2, 2p(1-p),$$ and $$p^2,$$ respectively.
Here is a plot of this mixture distribution (CDF) $$F.$$ I use a semi-log scale because there's a fairly large range of relevant probabilities:
The three Normal components centered at $$0,-10,-20$$ are apparent: these are the locations very close to the modes (where the slope of this plot is locally steepest). The red line shows the value $$1\% = 0.01.$$ The solution you seek is the value $$z$$ shown by the vertical gray line, situated where the red line intersects the graph.
Evidently, this solution is the zero of the function $$z\to F(z)-1/100.$$ Find it using any good univariate root finder. With double precision arithmetic you should obtain $$z \approx -9.8006135477.$$
This approach extends in an obvious manner to finding and working with distributions of the sums of any finite number of finite mixtures: it comes down to adding any pair of mixture components (such as $$X$$ and $$Y$$) and adding the discrete mixing variables (in this case, adding two iid copies of $$U$$).