Higher Engineering Mathematics

Ess-For-H8152.tex 19/7/2006 18: 2 Page 718

718 ESSENTIAL FORMULAE

J 0 (x)= 1 −

x^2

22 (1!)^2

+

x^4

24 (2!)^2

−

x^6

26 (3!)^2

+···

and J 1 (x)=

x

2

−

x^3

23 (1!)(2!)

+

x^5

25 (2!)(3!)

−

x^7

27 (3!)(4!)

+···

Legendre’s equation

The solution of (1−x^2 )

d^2 y

dx^2

− 2 x

dy

dx

+k(k+1)y= 0

is:

y=a 0

{

1 −

k(k+1)

2!

x^2

+

k(k+1)(k−2)(k+3)

4!

x^4 −···

}

+a 1

{

x−

(k−1)(k+2)

3!

x^3

+

(k−1)(k−3)(k+2)(k+4)

5!

x^5 −···

}

Rodrigue’s formula

Pn(x)=

1

2 nn!

dn(x^2 −1)n

dxn

Statistics and Probability

Mean, median, mode and standard deviation

Ifx=variate andf=frequency then:

mean ̄x=

∑

fx

∑

f

Themedianis the middle term of a ranked set of

data.

Themodeis the most commonly occurring value in

a set of data.

Standard deviation

σ=

√

√

√

√

[∑{

f(x− ̄x)^2

}

∑

f

]

for a population

Binomial probability distribution

If n=number in sample, p=probability of the

occurrence of an event and q= 1 −p, then the

probability of 0, 1, 2, 3,...occurrences is given by:

qn, nqn−^1 p,

n(n−1)

2!

qn−^2 p^2 ,

n(n−1)(n−2)

3!

qn−^3 p^3 ,...

(i.e. successive terms of the (q+p)nexpansion).

Normal approximation to a binomial distribution:

Mean=np Standard deviationσ=

√

(npq)

Poisson distribution

Ifλis the expectation of the occurrence of an event

then the probability of 0, 1, 2, 3,...occurrences is

given by:

e−λ, λe−λ, λ^2

e−λ

2!

, λ^3

e−λ

3!

,...

Product-moment formula for the linear correlation

coefficient

Coefficient of correlationr=

∑

xy

√[(∑

x^2

)(∑

y^2

)]

where x=X−X and y=Y−Y and (X 1 ,Y 1 ),

(X 2 ,Y 2 ),...denote a random sample from a bivari-

ate normal distribution andXandYare the means

of theXandYvalues respectively.

Normal probability distribution

Partial areas under the standardized normal curve —

see Table 58.1 on page 561.

Student’stdistribution

Percentile values (tp) for Student’stdistribution with

νdegrees of freedom — see Table 61.2 on page 587.