Suppose that f(x, t) is the probability of getting x suc-cesses during a time interval of length t when (i) the probability of a success during a very small time intervalfrom t to t + t is α · t, (ii) the probability of more thanone success during such a time interval is negligible, and (iii) the probability of a success during such a time inter-val does not depend on what happened prior to time t. (a) Show that under these conditions d[f(x, t)]dt = α[f(x − 1, t) − f(x, t)] (b) Show by direct substitution that a solution of thisinfinite system of differential equations (there is one foreach value of x) is given by the Poisson distribution withλ = αt.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
cesses during a time interval of length t when (i) the
from t to t + t is α · t, (ii) the probability of more than
one success during such a time interval is negligible, and
val does not depend on what happened prior to time t.
dt = α[f(x − 1, t) − f(x, t)]
infinite system of differential equations (there is one for
each value of x) is given by the Poisson distribution with
λ = αt.
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