Sums of Products of Two Normal Variables

Suppose that $X_1 ,\ldots,X_n,Y_1,\ldots,Y_n$ are all independent normal random variables with different means and variances. What is the PDF of the following random variable?

$$X_1Y_1+\cdots+X_nY_n$$

or is there any way that I can find an approximation for its PDF?

• I see no reason to expect a nice formula. Do you have one? – Did Sep 5 '13 at 14:39
• Then this is the prototypical question without answer, and opening a bounty will not change this. // @upvoters Why the upvote? – Did Sep 7 '13 at 9:21
• The best approximation is probably to invoke the CLT (if $n$ is large enough) to say that the whole thing will be normally distributed. Then all you have to know is the mean and variance. The mean you can certainly get easily. I'd have to think a bit about if you could get a formula for the variance in terms of the means and variances of the $X$ and $Y$ variables. – rajb245 Sep 7 '13 at 23:52
• There are generalizations of the CLT that apply even without identical distributions. Intuitively (forgetting the proofs), the sum of lots of random things happening tends towards normality, regardless of the underlying distributions, for many (most) practical problems in the real world. There are of course cases when this isn't true, but I suspect it is for your scenario. You can conduct some numerical experiments to convince yourself of this. I can provide some MATLAB code to motivate this, it's about ~15 lines. In my numerical investigations, even for $n=2$ its approximately normal. – rajb245 Sep 9 '13 at 3:46
• pastebin.com/Pqi5fEu4 – rajb245 Sep 10 '13 at 16:17

You want to know the distribution of

$$Z = \sum_{i=1}^n X_iY_i,$$

where $X_i$ and $Y_i$ are all normal and independent but not identically distributed. Let's calculate the mean and variance of $Z$:

$$\mathrm E[Z] = \sum_{i=1}^n \mathrm E[X_iY_i] = \sum_{i=1}^n \mathrm E[X_i]\mathrm E[Y_i].$$

You know the individual means of the normals, so the above sum can be calculated. Now the variance, knowing that covariance terms drop out because of uncorrelation due to independence

$$\mathrm{Var}[Z] = \sum_{i=1}^n \mathrm{Var}[X_iY_i] = \sum_{i=1}^n \mathrm E[Y_i]^2\mathrm{Var}[X_i]+\mathrm E[X_i]^2\mathrm{Var}[Y_i]+\mathrm{Var}[X_i]\mathrm{Var}[Y_i]$$

Again, this is all in terms of quantities you know, the means and variances of the underlying distributions.

So, IF the CLT applies, your answer is that $Z\sim\mathcal{N}(\mathrm{E}[Z],\mathrm{Var}[Z])$. Rigorously proving that this is true for your particular case and your particular distributions is up to you, but my claim is that it is true. I pastebinned some numerical calculations to support my claim.

• Updated MATLAB to demonstrate that the variance formula works too: pastebin.com/d6g5rs1A – rajb245 Sep 10 '13 at 16:46
• I have a question about your MATLAB code. Lets say N=2, Ntrials=2, so we have X=[a b;c d], Y=[e f;g h], to make samples for X1Y1+X2Y2 you just multiply a by e, I think we need to consider all the cases ,i.e. we should also multiply a by f too, the same for the summation. so with 2 number of trials and N=2, you create 2 samples, but the total number of possible cases is 16. – May Sep 22 '13 at 20:32
• $ae$ represents a single realization of the random variable $X_1Y_1$. $bf$ is another realization. Every time you want a realization of $X_1Y_1$, you generate two fresh random numbers and multiply them together. In fact, reusing them for the different possible "cases" as you suggest would lead to non-independent answers. Do you see how $ae$ and $af$ depend on each other? On the other hand, $ae$ and $bf$ do not. NEVER reuse random variables if you want independence. If you want independence, draw fresh numbers from the random number generators. – rajb245 Sep 25 '13 at 14:27
• Thanks a lot for your explanation, you are right. Is there any theory/book regarding this issue (i.e. creating dependent/independent random numbers)? – May Sep 25 '13 at 21:22

The product $Z_i=X_iY_i$ obeys the product normal distribution, which has a characteristic function given in: Characteristic function of product of normal random variables. As a result the characteristic function of your random variable is given by a product of the similar characteristic function (after matching to the various means and standard deviations). You can solve the question by approximating the Fourier transform of the resulting characteristic function.

• how can I use FT to approximate it? – May Sep 13 '13 at 17:12
• Sample the Fourier transform at the resolution you wish, while neglecting the tale of the characteristic function. If for example, all the expectations are 0 and all standard deviations are 1, start sampling the Fourier transform at the range $\omega \in [-10,10]] every 0.01 Hz and evaluate the PDF by performing a numeric Fourier transform over the samples. If you are using matlab I can help you write an example. – SBM Sep 14 '13 at 19:16 • Yes I am using MATLAB. that would be perfect thanks. – May Sep 14 '13 at 19:36 • The code assumes normalized random variables ($EX_i=0 \sigma_X=\sigma_Y=1$) and gives you two options of computations, numeric and analytic using the Matlab symbolic functions: N=5;$ $syms W X;$ $CF= (1/sqrt(1+W^2))^N;$ $minFreq=-10;$ $maxFreq=10;$ $resolutionFreq=0.01;$ $w=minFreq:resolutionFreq:maxFreq;$ $cf=double(subs(CF,W,w));$ $minX=-10;$ $maxX=100;$ $resolutionX=0.1;$ $x=minX:resolutionX:maxX;$ $% First Option$ $E=exp(jx'*w);$ $pdf=Ecf.'/(2*pi);$ $plot(x,pdf)$  $%Second Option$ $pdf=(ifourier(CF,W,X))';$ $pdf % Show the pdf as an expression$ $ezplot(pdf)$ \$ – SBM Sep 14 '13 at 20:46
• Thank you. But the characteristic function in (math.stackexchange.com/questions/74013/…) is for standard normal random variables. I think first I need to find the characteristic function for the general case. Any help would be appreciated, thank you. – May Sep 16 '13 at 23:14