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I am reading a text and I have a somewhat "abstract" question regarding probability. It involves a technique I have never seen before. The text is: Mobile Communication Systems by Parsons, page 208-209. Another text which contains the same information but in a slightly different form is Microwave Mobile Communications by Jakes, page 406-408.

The context is signal levels - a stochastic process can be either above or below a threshold, and we want to calculate probabilities related to this. There exist 2 input signals, $r_1(t)$ and $r_2(t)$. The output signal $R(t)$, takes on the value $r_i(t)$ until it falls below a threshold A, at which time it immediately takes on the value $r_j(t)$, $i \neq j$. $R(t)$ switches only when the current branch $r_i(t)$ falls below a threshold level A. Thus, it is entirely possible for R(t) to take on values below A, if both $r_1(t)$ and $r_2(t)$ are both below A. $r_1(t)$ and $r_2(t)$ have identically independent stationary probability distributions. So, we can calculate the probability of $r_i(t)$ at any time being below a level, within a range, or above a level.

Suppose the probability $P(r_i(t) < A) = q$. Suppose the average amount of time $r_i(t)$ is above level A is $\tau_p$. The average amount of time is defined as: the total amount of time above the level divided by the number of times the signal crosses into the level. Suppose the average amount of time $r_i$ is below level A is $\tau_q$. A "successful" transition is defined as: the switch occurs such that the signal that is being switched into is already above the level A. A "unsuccessful" transition is defined as: the switch occurs such that the signal that is being switched into is below the level A (so we did not improve our situation - a higher signal level is considered better).

We also have the following conditional probability distributions: $P(r_i(t) < B | r_i(t) > A) = x, B>A$ $P(r_i(t) < B | r_i(t) > A) = 0, B $P(r_i(t) < B | r_i(t) < A) = 1, A $P(r_i(t) < B | r_i(t) < A) = y, A>B$

We are interested in calculating the probability $P(R(t) < B)$ The statement (paraphrased) that makes me confused is: To obtain the overall cumulative distribution, it is necessary to combine the four equations above, with a weighting to account for the relative times over which they apply.

The book goes on to say the average duration of a segment following a successful transition is $\frac{\tau_p}{2}$. The average duration of a segment following an unsuccessful transition is $\frac{\tau_q}{2} + \tau_p$. I can agree with the last two sentences.

Then, my question is, why is this statement true, exact quote from the book: "The probabilities of successful and unsuccessful transitions are (1-q) and q, respectively, and the weighting appropriate to each distribution is proportional to probability of occurrence X average duration". Why is this true? I am looking at the units here, and we are multiplying a time by a probability. Eventually the final result $P(R(t) < B)$ is the product of a 2x2 matrix (consisting of the four conditional probabilities) by a vector. The vector contains the "probability of occurrence X average duration", normalized (the vector elements sum to 1). Why is this true?

I know this question is (too) long. Thanks for any help, I will attempt to clarify if I can.

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    I asked my pro$f$essor, and he said it has something to do with ergodicity. I have narrowed the problem down to something like $P(R(t) < B) = P(R(t) < B | R(t) < A)unknown$f$actor1 + P(R(t) < B | R(t) > A)unknownfactor2$. It turns out the unknown factors "correspond" to the avera$g$e length o$f$ the time segment multiplied by the probability reaching that time segment. The question is what is the mathematical de$f$inition of "correspond"? Here we are sayin$g$ corresponds means probability X average_time_segment.2011-04-06

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