Binomial Distribution Part 5 deals with fitting & familiaring some concepts of B D under the complementary Statistics syllabus of University of Calicut in BSc core of Mathematics, Physics & Computer Science.

1. Welcome
Suchithra’s Statistics Classes
Suchithra's Statistics Classes -- Binomial Distribution, Part 5

2. Binomial Distribution – B.D
Part – 5
(Based on complementary Statistics
of Bsc , University of Calicut)
Suchithra's Statistics Classes -- Binomial Distribution, Part 5
Fitting of B D & Some Problems

3. Let n independent trials of a Bernoulli sequence constitute one
experiment and let this experiment be repeated N times.
Then we can classify the N experiments into experiments with 0
successes, experiments with 1 success, ----- experiment with n
successes. The probabilities corresponding to 0,1,2,….,n &
Expected frequencies are obtained as follows.
X 0 1 2 .. x .... n X
f(x) qn (nC1)pqn-1 (nC2) p2qn-2 .. (nCx) pxqn-x .... pn f(x)
*N.f(x) N.f(0) N.f(1) N.f(2) .. N.f(x) .... Nf(n) *N.f(x)
*N.f(x) =Expected No: of experiments
“p” is obtained from Mean = = np
Fitting of B D.
Suchithra's Statistics Classes -- Binomial Distribution, Part 5

4. Seven coins are tossed and number of heads noted, the
experiment is repeated 130 times and the following distribution
is obtained
No. of heads 0 1 2 3 4 5 6 7 Total
Frequencies 7 6 19 35 30 23 7 3 130
Suchithra's Statistics Classes -- Binomial Distribution, Part 5
Fit a Binomial distribution assuming
1) The coins are unbiased
2) The nature of the coins is not known

5. Suchithra's Statistics Classes -- Binomial Distribution, Part 5
Case I.
When the coin is unbiased p = q = 1/2 = 0.5,here n=7
x Expected frequencies =130 f(x)
0 1.023425≈ 1
1 7.109375≈ 7
2 21.328125≈ 21
3 35.546875≈ 36
4 35.546875≈ 36
5 21.328125≈ 21
6 7.109375≈ 7
7 1.023425≈ 1
Total 1 130

6. Suchithra's Statistics Classes -- Binomial Distribution, Part 5
Case 2
If the nature of the coins is not known ,calculate p from the mean
of the data .
In B D mean = np,; p= np/n ;find q= 1-p, & n=7.
No. of
heads
Frequencies
f
fx
0 7 0
1 6 6
2 19 38
3 35 105
4 30 120
5 23 115
6 7 42
7 3 21
Total 130 449
Then Proceed like case 1

7. Suchithra's Statistics Classes -- Binomial Distribution, Part 5
Comment on the statement “ the mean of the B D is 10 and
its std.deviation is 4”
& Var = npq= 16
So the statement is false
Mean = np = 10 & std.deviation = 4

8. Suchithra's Statistics Classes -- Binomial Distribution, Part 5
since X & Y are independent.
.

9. Suchithra's Statistics Classes -- Binomial Distribution, Part 5

10. Suchithra's Statistics Classes -- Binomial Distribution, Part 5
Suppose the mgf of a r.v X is of the form
What is the mgf of the r.v Y=3X+2 ? Evaluate E(X) & V(X).
Comparing we get q=0.6, p= 0.4 & n=8
E(X)= np = 8 x 0.4 =3.2
V(X) = npq = 8 x 0.4 x 0.6 =1.92

11. X follows the B D with parameters n=6 & p,
If 9 P(4) = P(2) find p.
Suchithra's Statistics Classes -- Binomial Distribution, Part 5

12. The mean and variance of a binomial variate X are 16 & 8.
Find P(X=0) & P(X>=2)
Suchithra's Statistics Classes -- Binomial Distribution, Part 5

13. Measures of B D at a glance
Pmf/pdf is P(X=x) = f(x) = nCx px qn-x
Suchithra's Statistics Classes -- Binomial Distribution, Part 5

14. The mode corresponds to the values of x which lies between
.
Suchithra's Statistics Classes -- Binomial Distribution, Part 5
The recurrence relation between binomial probabilities

15. Things to be known from this class
• How to fit a B D
• Understanding the measures of BD
i. Try to form mathematical model using given
information.
ii. Calculate n & p
iii. Using these n & p calculate the necessary
characteristics
Suchithra's Statistics Classes -- Binomial
Distribution, Part 5
Materials are adopted from the reference books prescribed by the syllabus

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Suchithra’s Statistics classes
Suchithra's Statistics Classes -- Binomial Distribution, Part 5