|1.||x and y both normally distributed|
|2.||x and y independent|
|3.||standard deviation of x = standard deviation of y. Let the joint standard deviation be σ|
|4.||mean of x = mean of y = 0.|
That is, we are assuming a 'circular' normal distribution of errors around the target, and no 'bias'.
Let p(x) be the probability distribution of x, p(y) the probability distribution of y, and p(x,y) their joint probability distribution. Then
To obtain the single-shot kill probability we need to find the probability of the bomb exploding within the 'lethal radius' of its target. Simplifying some complex issues, a dimensional argument establishes that the lethal radius rk is a function of the one-third power of bomb's yield y. It is also inversely related to the hardness h of the target i.e. the degree of pounds per square inch overpressure the target can withstand. The relation is mathematically