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from 15. We see that the “width” of the curve in Eig. 6-2, measured from the
center, is just about 3 units, in agreement with this result.
W© are now in a position to consider a question we have avoided until now.
How shall we tell whether a coin is “honest” or “loaded”? We can give now at
least a partial answer. EFor an honest coin, we expect the fraction of the times
heads appears to be 0.5, that is,
———— = 035. 6.13
_ (6.13)
W© aÏso expect an actual Ấy to deviate from Ñ/2 by about v N/2, or the ƒfraction
to deviate by
1LvN 1
N 2 2vN.
The larger is, the closer we ezpect the fractlon Wg/N to be to one-half.
In Eig. 6-6 we have plotted the fraction Vy/N for the coïin tosses reported ear-
lier in this chapter. We see the tendency for the fraction of heads to approach 0.5
for large W. Unfortunately, for any given run or combination of runs there is no
guarantee that the observed deviation will be even øcør the ezpected deviation.
'There is always the finite chance that a large fuctuation—a long string of heads
or tails—will give an arbitrarily large deviation. All we can say is that jƒ the
deviation is near the expected 1/2VWN (say within a factor of 2 or 3), we have
no reason to suspect the honesty of the coin. lÝ it is much larger, we may be
suspicious, but cannot prove, that the coïn is loaded (or that the 6osser is cleverl).
--- Trang 130 ---
1.0
FRACTION
HEADS Ö~?
0.5 <
0 1 2 4 8 16 32 64 128 256 512 1024 2048 4096
N (COIN TOSSES)
Fig. 6-6. The fraction of the tosses that gave heads in a particular
sequence of Ñ tosses of a penny.
W©e have also not considered how we should treat the case of a “coin” or
sơme similar “chancy” object (say a stone that always lands in either of two
positions) that we have good reason to believe should have a diferent probability
for heads and tails. We have deined P(H) = (Nn)/N. How shall we know what
to ezpec£ for N„? In some cases, the best we can do is to observe the number
of heads obtained in large numbers of tosses. For want of anything better, we
must set (N;;) = Nư(observed). (How could we expect anything else?) We must
understand, however, that in such a case a diferent experiment, or a diferent
observer, might conclude that P(H) was diferent. We would ezpect, however,
that the various answers should agree within the deviation 1/2VN [if P(H) is
near one-half]. An experimental physicist usually says that an “experimentally
determined” probability has an “error,” and writes
P(H) N + 2VN (6.14)
'There is an implication in such an expression that there 7s a “true” or “correc$”
probability which couwld be computed If we knew enouph, and that the observation
may be in “error” due to a Ñuctuation. 'There is, however, no way to make such
thinking logically consistent. It is probably better to realize that the probability
concept is in a sense subjective, that it is always based on uncertain knowledge,
and that its quantitative evaluation is subject to change as we obtain more
Information.
--- Trang 131 ---
6-4 A probability distribution
Let us return now to the random walk and consider a modification of it.
Suppose that in addition to a random choïce of the đecføn (+ or —) of each
step, the /ength of each step also varied in some unpredictable way, the only
condition being that on the auerage the step length was one unit. 'Phis case is
more representative of something like the thermal motion of a molecule in a gas.
Tí we call the length of a step Š, then Š may have any value at all, but most often
will be “near” 1. To be specifc, we shall let (S2) = 1 or, equivalently, Sz„;„ = 1.
Our derivation for (D?) would proceed as before except that Eq. (6.8) would be
changed now to read
(DẬ) = (DẶ +) + (52) = (DẶ_ ¡) +1. (6.15)
W© have, as before, that
(DV})=N. (6.16)
'What would we expect now for the distribution of distances D? What is, for
example, the probability that J2 = 0 after 30 steps? The answer is zerol 'Phe
probability is zero that D will be amw particular value, since there is no chance at
all that the sum of the backward steps (of varying lengths) would exactly equal
the sum oŸ forward steps. We cannot plot a graph like that of Eig. 6-2.
W© can, however, obtain a representation similar to that of Fig. 6-2, if we ask,
not what is the probability of obtaining D exactly equal to 0, 1, or 2, but instead
what is the probability of obtaining D near 0, 1, or 2. Let us define P(z, Az)
as the probability that D will lie in the interval Az located at z (say from ø
to z-+ Az). We expect that for small Az+ the chance of DĐ landing in the interval
is proportional to Az, the width of the interval. So we can write
Pí(œ, Az) = p(œ) Az. (6.17)
The function ø(x) is called the przobabilitụ densitg.
The form oŸ p(+) will depend on , the number of steps taken, and also on the
distribution of individual step lengths. We cannot demonstrate the proofs here,
but for large W, p(#) is the sarme for all reasonable distributions in individual
step lengths, and depends only on ÑW. W© plot (+) for three values oŸ Ý in
Fig. 6-7. You will notice that the “halfwidths” (typical spread from # = 0) of
these curves is v(, as we have shown it should be.
--- Trang 132 ---
PROBABILITY DENSITY
N = 10,000 STEPS
40,000 STEPS
160,000 STEPS
—700 —600 —500—400—300—200-100 0 100 200 300 400 500 600. 700
D = DISTANCE FROM START
Fig. 6-7. The probability density for ending up at the distance 2 from
the starting place in a random walk of N steps. (D is measured in units
of the rms step length.)
You may notice also that the value oŸ ø0() near zero is inversely proportional
to VN. This comes about because the curves are all of a similar shape and theïr
areas under the curves must all be equal. Since ø(#) Az is the probability of
fñnding Din Az when Az is small, we can determine the chance of finding D
sơmcuhere inside an arbitrary interval from # to #a, by cutting the interval in
a number of small increments Az and evaluating the sum of the terms ø() Az