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)