# Flying Bomber aircraft through SAM sites - combining normal distributions

I have been puzzling over this for days but I don't think this was covered at school

We simultaneously fly a known number of bomber aircraft, K, through three sequential batteries of surface to air missile (SAM) batteries, A B C.

Assume the planes are incapable of destroying the SAM batteries.

Assume that the SAM batteries are sufficiently spaced out such that only one battery A,B or C can engage the planes at once. I.e. the enemy gets three separate goes, once with each battery to shoot at the planes.

E.g if we start with 10 planes, battery A shoots down 3, B then gets a go at the remaining 7 it shoots down 2 etc etc...

Assume each SAM battery shoots down a random number of planes. This random number is drawn from a normal distribution unique to each battery i.e. Mu_A, Var_A, Mu_B, Var_B etc.

How would one combine these three distributions into a single distribution from which could be drawn the number of planes that make it through all three layers? I need the parameters i.e. Mu_Combined, Var_Combined of the combined distribution

Thanks very much

EDIT:

Use of the normal distribution is essential. In the model I am using the number of planes shot down is not particularly dependent on the number there are to shoot as it assumed each battery gets one shot and can only shoot some integer number of missiles. In addition the model uses a continuous distribution with the same mu and var as the binomial. Apparently it is a well established approximation whose accuracy can be improved using a continuity correction.(applied later)

• "Stops a random number of projectiles" could mean various things. Could you explain this? (Asking us to assume a blatant counterfactual such as "sheets ... are not damaged" when they are perforated makes this question particularly hard to understand! If you are using the sheets and projectiles as a metaphor for something else, you might be better off explaining what you're really studying.) – whuber Jan 4 '13 at 21:38
• @whuber Description amended – RNs_Ghost Jan 4 '13 at 22:37

The sum of a linear combination of independent normal variables is also normally distributed, with mean equal to the sum of the linear combination of the means and variance the sum of the variances times the linear coefficients squared. So the distribution of planes getting through, given your comments on my original answer, must be have a normal distribution with mean:

$N-\mu_A-\mu_B-\mu_C$

and variance:

$(-1)^2Var_A+(-1)^2Var_B+(-1)^2Var_C$

If there is any dependence between thre three SAM batteries' performance - eg on particular weather conditions, all batteries have a bad day - you can take this into account too extending to three variables the standard results for sums of dependent random variables: $Var(X_1+X_2)=Var(X_1) + Var(X_2) +2Cov(X_1X_2)$

My scepticism about this as a model remains - certainly if your numerical example is at all representative of the sort of numbers involved - so I leave my original answer below. If your numerical example is representative, the number of planes getting through would be normally distributed with a mean of around 2 or 3 and a variance of about 150 - 40% of the distribution in the (impossible) negative zone, and another 25% indicating more planes get through than the ten that started! So this might be why it seemed to you that it should be more complicated than just the sum of normally distributed variables.

I think the problem is misconceived because of the paragraph:

Assume each SAM battery shoots down a random number of planes. This random number is drawn from a normal distribution unique to each battery i.e. Mu_A, Var_A, Mu_B, Var_B etc.

It is not possible for the number of planes shot down to be exactly normally distributed, because the number shot down has to be an integer, and a normal distribution is continuous. Further, unless the number of planes going through is very large and the number shot down relatively small, it seems unlikely that the number shot down will be even approximately normally distributed. Finally and most importantly, surely the number shot down is not going to be characterised just by SAM-specific parameters (Mu_A, Var_A) etc but also by the number of planes going through.

A more plausible (but still very simplifying) model would be that each SAM battery shoots down a random number of planes from a binomial distribution, characterised by $p_A$, $p_B$, etc (SAM-specific parameters) and $N$ - the number of planes that reach that battery. I say this is still inadequate because there would be diseconomies of scale - if the SAM is swamped with 1000 planes at once presumably the proportion it hits will be less than if it only has 10 or so to focus on. But for your problem (judging from your numerical example), I would say this is a much better way of looking at than as normally distributed.

Reconceived this way, the chance any single plane has of getting through all three SAM batteries (assuming - again implausibly because of pilot and plane skill etc, but maybe ok for now - that these are three independent events) is

$(1-p_A)(1-p_B)(1-p_C)$

You could use this fact to derive the actual distribution you are interested in of the number of planes that go through (I won't do this for you as it looks like homework).

Now, if I've missed the point and the "normal distribution" is an essential part of the problem, everything I've written above is unhelpful. But the normal distribution approach could only work if the number of planes shot down is not particularly dependent on the number there are to shoot.

• Thanks @Peter Ellis! I'm afraid the normal distribution is essential. In the model I am using the number of planes shot down is not particularly dependent on the number there are to shoot as it assumed each battery gets one shot and can only shoot some integer number of missiles. In addition the model uses a continuous distribution with the same mu and var as the binomial. Apparently it is a well established approximation whose accuracy can be improved using a continuity correction.(applied later) – RNs_Ghost Jan 4 '13 at 23:38
• In addition this is not homework (i wish my homework was this cool) it is the Salvo model of missile combat: A Stochastic Salvo Model for Naval Surface Combat Author(s): Michael J. Armstrong Reviewed work(s): Source: Operations Research, Vol. 53, No. 5 (Sep. - Oct., 2005), pp. 830-841 – RNs_Ghost Jan 4 '13 at 23:41
• A better link is to stats.stackexchange.com/questions/19948/… – Peter Ellis Jan 5 '13 at 0:14
• Thanks very much. There are lots of details of the model not detailed here and lots of application papers that apply the model to real battles and the same concerns are brought up and addresses. – RNs_Ghost Jan 5 '13 at 0:17
• @RRs_Ghost : suggested reading Alan Washburn, Moshe Kress. Combat modeling. Springer 2009. ISBN 978-1-4419-0789-9. – Deer Hunter Jan 5 '13 at 6:34