Wikipedia:Reference desk/Archives/Mathematics/2017 May 25

= May 25 =

Probability Question about Rectangular Pulses
Given an interval $$I=[0,t]$$, and given two pulse waves (rectangular waves), $$f,g$$,and we randomly (uniform distribution on the interval between 0 and $$f$$'s wavelength, $$\lambda_f$$) choose a phase shift $$r$$ to the pulse $$f$$. What is the probability $$\Pr_{r\in U[0,\lambda_f]}(\exists x\in I:f(x+r)=1, g(x)=1)$$? 212.179.21.194 (talk) 08:55, 25 May 2017 (UTC)
 * Not quite clear on the question. Are the periods of f and g meant to be the same? Also, since they don't appear in the formula, where do t and I come in. It seems likely what you're asking is equivalent to the probability that two arcs of circle (maybe subsegments of a segment) intersect, but it's hard to tell how the circle radius and the arcs are selected. --RDBury (talk) 03:57, 26 May 2017 (UTC)
 * I do not assume that the periods of f and g are the same. I expect the answer (the probability) to be a function of their periods. In addition, the greater t and I are, the higher probability there is as we have more points of f and g, so it's more likely to find some point x that satisfies the desired. 213.8.204.59 (talk) 09:16, 26 May 2017 (UTC)
 * You mention the random phase shift of f, but what about g? Is it randomly shifted vs. the origin, or is it synced with 0?  (This becomes less important for large t.)  You do not mention the duty cycle of f and g, and presumably want an answer which is a function of f and g's periods and duty cycles as well as t.
 * I doubt that a simple formula is possible. When f and g are not in resonance, the probability will approach 1 with large t, but with resonance it will approach that of the overlap probability during a single resonance period.  That sounds messy. -- ToE 10:04, 26 May 2017 (UTC)