I Can Continuous Approximation Improve Understanding of 1D Random Walks?

AI Thread Summary
In the discussion about continuous approximation in 1D random walks, it is noted that for large total steps (N), the binomial probability distribution exhibits a maximum at a specific value of steps to the right (n1). The fractional change in the probability W when n1 increases by one is small, allowing W to be treated as a continuous function despite its discrete nature. This approximation facilitates the use of calculus to find the maximum of W by setting its derivative to zero. Questions arise regarding the assignment of values to W between integral points, with references made to the normal approximation of the binomial distribution. The conversation emphasizes the utility of continuous functions in simplifying calculations in probabilistic models.
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Reif,pg 14. ##n_1## is the number of steps to the right in a 1D random walk. ##N## are the total number of steps

"When ##N## is large, the binomial probability distribution ##W\left(n_{1}\right)##
##W\left(n_{1}\right)=\frac{N !}{n_{1} !\left(N-n_{1}\right) !} p^{n_{1}} q^{N-n_{1}}##
tends to exhibit a pronounced maximum at some value ##n_{1}=n_{1}##, and to decrease rapidly as one goes away from ##\tilde{n}_{1}##. Let us exploit this fact to find an approximate expression for ##W\left(n_{1}\right)## valid when ##N## is sufficiently large.
If ##N## is large and we consider regions near the maximum of ##W## where ##n_{1}## is also large, the fractional change in ##W## when ##n_{1}## changes by unity is relatively quite small, i.e.,
##\left|W\left(n_{1}+1\right)-W\left(n_{1}\right)\right| \ll W\left(n_{1}\right)##
Thus ##W## can, to good approximation, **be considered as a continuous function of the continuous variable ##n_{1}##**, although only integral values of ##n_{1}## are of physical relevance. The location ##n_{1}=\tilde{n}## of the maximum of ##W## is then approximately determined by the condition ##\frac{d W}{d n_{1}}=0 \quad##"

* I'm not able to understand why
##
\left|W\left(n_{1}+1\right)-W\left(n_{1}\right)\right| \ll W\left(n_{1}\right)
##means we can use a continuous approximation.

* How do we approximate a discrete function by a continuous one. Since ##W## has values defined only at integral values, what values do we assign to the continuous function between any two consecutive integers i.e what value does ##W(0.5)## have in the continuous approximation?
 
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I expect the author is referring to the normal approximation to the binomial distribution. You fit it by specifying the mean and variance of the normal distribution to equal the mean (np) and variance (npq) of the binomial. You can then calculate the PDF or CDF at any value rather than just integer values.
 
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