1. BINOMIAL DISTRIBUTION
The context
The properties
Notation
Formula
Mean and variance

2. BINOMIAL DISTRIBUTION THE CONTEXT
An important property of the binomial distribution:
An outcome of an experiment is classified into one of two mutually exclusive categories - success or failure.
Example: Suppose that a production lot contains 100 items. The producer and a buyer agree that if at most 2 out of a sample of 10 items are defective, then all the remaining 90 items in the production lot will be purchased without further testing. Note that each item can be defective or non defective which are two mutually exclusive outcomes of testing. Given the probability that an item is defective, what is the probability that the 90 items will be purchased without further testing?

3. BINOMIAL DISTRIBUTION THE CONTEXT
Trial Two Mut. Excl. and exhaustive outcomes
Flip a coin Head / Tail
Apply for a job Get the job / not get the job
Answer a Multiple Correct / Incorrect
choice question

4. BINOMIAL DISTRIBUTION THE PROPERTIES
The binomial distribution has the following properties:
1. The experiment consists of a finite number of trials. The number of trials is denoted by n.
2. An outcome of an experiment is classified into one of two mutually exclusive categories - success or failure.
3. The probability of success stays the same for each trial. The probability of success is denoted by
4. The trials are independent.

5. BINOMIAL DISTRIBUTION THE NOTATION
n : the number of trials
r : the number of observed successes
: the probability of success on each trial
Note:
Since success and failure are two mutually exclusive and exhaustive events
The probability of failure on each trial is 1-

6. BINOMIAL DISTRIBUTION THE PROBABILITY DISTRIBUTION
The binomial probability distribution gives the probability of getting exactly r successes out of a total of n trials.
The probability of getting exactly r successes out of a total of n trials is as follows:
Note: In the above gives the number of different ways of choosing r objects out of a total of n objects

7. Example 1
There are 5 parts to be checked. We are considering exactly r=2 defectives occurring in those parts. (see page 208). (i) How many ways two defectives can be found?
(ii) Since the elementary event in each of these cases has probability , so what is the probability of getting exactly two defectives.

8. Example from page 209

9. BINOMIAL DISTRIBUTION MEAN AND VARIANCE
If X is a binomial random variable, the mean and the variance of X are:
E(R) is the mean or expected value of R
V(R) is the variance of R
n is the number of trials
is the probability of success on each trial
See book page 210 for the derivation for E(R)

10. Cumulative Probability and the Binomial Probability Table
When n is small, it is easy to compute binomial probabilities using a calculator
To ease the computational burden, the more common binomial probabilities have been tabled.
Such tables are usually constructed in terms of cumulative probabilities
A cumulative probability is found by summing the individual probability terms applicable to all levels of the random variable that fall at or below a specified point.

11. Cumulative Probability and the Binomial Probability Table
The resulting sums themselves form the probability distribution function, defined for discrete distributions in terms of the cumulative sum of the probability mass function.
The following expression applies.

12. See page 211, 212 and 214 for examples

13. The Normal Distribution
Most frequently encountered in statistics is the normal distribution, often referred to as the Gaussian distribution.
Every normal frequency curve is centered on the population mean and symmetric about this point

14. NORMAL DISTRIBUTION The probability density function
f(x), area and the effect of changing mean and variances
Given x, find probability
Given probability, find x

15. NORMAL DISTRIBUTION THE PROBABILITY DENSITY FUNCTION
If a random variable X with mean  and standard deviation  is normally distributed, then its probability density function is given by

16. The Normal distribution and the Population Frequency Curve
When plotted, the foregoing function takes the familiar “bell shape”.
The two parameters
entirely specify a particular normal curve.
Also, this location is both the median and standard deviation
It is obvious that the normal curve will never touch the x-axis, since f(x0 will be nonzero over the entire real line, from –infinity and positive infinity.

17. NORMAL DISTRIBUTION THE PROBABILITY DENSITY FUNCTION
0.0000
0.0100
0.0200
0.0300
0.0400
0.0500 20 40 60 80
100 x -
VALUES f ( )
Mean, m
=50
SD, s
=10
Area under
the curve
= 1.00
Area between
the vertical lines = P
(40
 60)

18. NORMAL DISTRIBUTION EFFECT OF CHANGING STANDARD DEVIATION
0.0000
0.0100
0.0200
0.0300
0.0400
0.0500 50
100
150
200
x-values
f(x)
SD,=10
SD,=15
SD,=20
Mean,= 100

19. NORMAL DISTRIBUTION EFFECT OF CHANGING MEAN
SD,s=10

20. STANDARD NORMAL DISTRIBUTION

21. STANDARD NORMAL DISTRIBUTION

22. STANDARD NORMAL DISTRIBUTION RELATIONSHIP BETWEEN x AND z
If a random variable X is normally distributed with mean  and standard deviation , then

23. STANDARD NORMAL DISTRIBUTION TABLE, z-VALUES, AREA AND PROBABILITY
Example 1.1: Table 3, Appendix B, p. 837 shows the area under the curve from z=0 to some positive z value. For example, the area from z=0 to z= =1.34 is