1000 Solved Problems in Modern Physics

32 1 Mathematical Physics

Fig. 1.5Soap film stretched
between two parallel circular
wires

1.2.13 StatisticalDistributions

1.93 Poisson distribution gives the probability thatxevents occur in unit time

when the mean rate of occurrence ism.

Px=

e−mmx

x!

(a) Show thatPxis normalized.

(b) Show that the mean rate of occurrence or the expectation value<x>,is

equal tom.

(c) Show that the S.D.,σ=

√

m

(d) Show thatPm− 1 =Pm

(e) Show thatPx− 1 =mxPmandPx+ 1 =xm+ 1 Px

1.94 The probability of obtainingxsuccesses inN-independent trials of an event

for whichpis the probability of success andqthe probability of failure in a

single trial is given by the Binomial distribution:

B(x)=

N!

x!(N−x)!

pxqN−x=CxNpxqN−x

(a) Show thatB(x) is normalized.

(b) Show that the mean value isNp

(c) Show that the S.D. is

√

Npq

1.95 A G.M. counter records 4,900 background counts in 100 min. With a radioac-

tive 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 whenpis held fixed, the Binomial distribution tends to a nor-

mal distribution asNis increased to infinity.

(b) IfNpis held fixed, then binomial distribution tends to Poisson distribu-

tion asNis increased to infinity.

1.97 The background counting rate isband background plus source isg.Ifthe

background is counted for the timetband the background plus source for a

timetg, show that if the total counting time is fixed, then for minimum sta-

tistical error in the calculated counting rate of the source(s),tbandtgshould

be chosen so thattb/tg=

√

b/g