This document is part 4 of Suchithra's statistics classes focusing on the binomial distribution, covering concepts such as the probability function, moment generating function, and characteristics of the distribution. Key topics include the additive property of binomial distributions, calculating and interpreting the mode, and deriving central moments. The document concludes with an invitation to like and subscribe if found useful.

1. Welcome
Suchithra’s Statistics Classes
Suchithra's Statistics Classes -- Binomial Distribution, Part 4

2. Binomial Distribution – B.D
Part – 4
(Based on complementary Statistics
of Bsc , University of Calicut)
Suchithra's Statistics Classes -- Binomial Distribution, Part 4

3. Suchithra's Statistics Classes -- Binomial Distribution, Part 4
P(X=x) = f(x) = nCx px qn-x
Moment generating function --- m.g.f

4. Suchithra's Statistics Classes -- Binomial Distribution, Part 4

5. Suchithra's Statistics Classes -- Binomial Distribution, Part 4

6. Suchithra's Statistics Classes -- Binomial Distribution, Part 4

7. Suchithra's Statistics Classes -- Binomial Distribution, Part 4
Additive property of B. D
This can be extend to any number of independent r.vs if p is same.
If the p is not the same then X+Y will not be binomial.

8. Suchithra's Statistics Classes -- Binomial Distribution, Part 4
Characteristic function of B.D

9. Suchithra's Statistics Classes -- Binomial Distribution, Part 4
Mode of a probability distribution is the value of the r.v
having maximum probability.
If x is the mode of a distribution, f(x) will be the maximum
This can be split into
Mode of B.D

10. Suchithra's Statistics Classes -- Binomial Distribution, Part 4

11. Suchithra's Statistics Classes -- Binomial Distribution, Part 4
On simplification we get
Therefore the mode corresponds to the values of x which lies
between
Combing (1) & (2)
If (n+1) p is an integer say k , then the mode is at X=k & X=k-1 ,i.e;
the B.D is bimodal.
If (n+1)p is not an integer, then integer part of (n+1)p is the mode of
the B.D (Uni model).

12. Things to be clear from this class
• How to find m.g.f of a B.D ? Hence deduce
central moments.
• Additive property of B.D.
• How to find the Characteristic function of a
B.D.
• Calculate & interpret Mode of a B.D.
Suchithra's Statistics Classes -- Binomial Distribution, Part 4

13. Thank you for watching,
if this is found to be useful then
like & subscribe.
Suchithra’s Statistics classes
Suchithra's Statistics Classes -- Binomial Distribution, Part 4