Paper 4: Fundamentals of Business Mathematics & Statistic

10.12 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Theoretical Distribution

Putting r = 0, 1, 2....n, we can get the expected or theoretical frequencies of the binomial distribution as

follows :

Number of Success(r) Expected or theoretical frequency

0

()

n

NP r

Nq

1 N nC 1 qn–1 p

2 N nC 2 qn–2 p^2

rN nCr qn–r pr

nN pn

Example 12 :

8 coins are tossed at a time, 256 times.. Find the expected frequencies of success (getting a. head) and

tabulate the result obtained.

Solution :

(^11) ; ; 8 : 256
pqn N==== 22
The probability of success r times in n -trials is given by nrnrCpqr −
(or) P(r) = nrnrCpqr −

=

118

22

rr

nCr

⎛⎞⎛⎞−

⎜⎜⎜⎜⎟⎟⎟⎟⎟⎟⎟⎟

⎜⎜⎝⎠⎝⎠

=

8 18

Cr 2

⎛⎞⎜ ⎟⎟

⎜⎜⎝⎠⎟⎟

Frequencies of 0, 1, 2, .... 8 successes are :

Success Expected Frequency

(^080)

256 1

256 C

⎛⎞⎜ × ⎟⎟

⎜⎜⎝⎠⎟⎟ 1

1 8 1

256 1

256 C

⎛⎞⎜ × ⎟⎟

⎜⎜⎝⎠⎟⎟ 8

(^282)

256 1

256 C

⎛⎞⎜ × ⎟⎟

⎜⎜⎝⎠⎟⎟ 28

(^383)

256 1

256 C

⎛⎞⎜ × ⎟⎟

⎜⎜⎝⎠⎟⎟ 56

4 8 4

256 1

256 C

⎛⎞⎜ × ⎟⎟

⎜⎜⎝⎠⎟⎟ 70