32
1 Mathematical Physics
Fig. 1.5 Soap film stretched
between two parallel circular
wires
1.2.13 Statistical Distributions
1.93 Poisson distribution gives the probability that x events occur in unit time
when the mean rate of occurrence is m.
P x =
e
−m m
x
x!
(a) Show that P x is normalized.
(b) Show that the mean rate of occurrence or the expectation value < x >, is
equal to m.
(c) Show that the S.D., σ =
√
m
(d) Show that P m−1 = P m
(e) Show that P x−1 =
x
m
P m and P x+1 =
m
x+1
P x
1.94 The probability of obtaining x successes in N -independent trials of an event
for which p is the probability of success and q the probability of failure in a
single trial is given by the Binomial distribution:
B(x) =
N !
x!(N − x)!
p
x q
N −x
= C
N
x p
x q
N −x
(a) Show that B(x) is normalized.
(b) Show that the mean value is N p
(c) Show that the S.D. is
√
N pq
1.95 A G.M. counter records 4,900 background counts in 100 min. With a radioactive source in position, the same total number of counts are recorded in
20 min. Calculate the percentage of S.D. with net counts due to the source.
[Osmania University 1964]
1.96 (a) Show that when p is held fixed, the Binomial distribution tends to a normal distribution as N is increased to infinity.
(b) If N p is held fixed, then binomial distribution tends to Poisson distribution as N is increased to infinity.
1.97 The background counting rate is b and background plus source is g. If the
background is counted for the time t b and the background plus source for a
time t g , show that if the total counting time is fixed, then for minimum statistical error in the calculated counting rate of the source(s), t b and t g should
be chosen so that t b /t g =
√
b/g