1000 Solved Problems in Modern Physics

1.1 Basic Concepts and Formulae 9

to have a stationary value (maximum or minimum). The integrand is taken to be

a function of the dependent variableyas well as the independent variablexand

y′=dy/dx. The limitsx 1 andx 2 are fixed and at each of the limitsyhas definite

value. The condition thatIshall be stationary is given by Euler’s equation

∂F

∂y

−

d

dx

∂F

∂y′

= 0 (1.57)

WhenFdoes not depend explicitly onx, then a different form of the above
equation is more useful

∂F

∂x

−

d

dx

(

F−y′

∂F

∂y′

)

= 0 (1.58)

which gives the result

F−y′

∂F

∂y′

=Constant (1.59)

Statistical distribution

Binomial distribution

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 distributionB(x).

B(x)=

N!

x!(N−x)!

pxqN−x=CxNpxqN−x (1.60)

B(x) is normalized, i.e.

∑N

x= 0 B(x)=^1 (1.61)

It is a discrete distribution.

The mean value,

〈x〉=Np (1.62)

The S.D.,

σ=

√

Npq (1.63)