BINOMIAL DISTRIBUTION

1
BINOMIAL DISTRIBUTION

• The context
• The properties
• Notation
• Formula
• Mean and variance

2
BINOMIAL DISTRIBUTIONTHE 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 DISTRIBUTIONTHE 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 DISTRIBUTIONTHE 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 DISTRIBUTIONTHE NOTATION

• 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 DISTRIBUTIONTHE 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 r2 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 .00729, so what is the
  probability of getting exactly two defectives.

8
Example from page 209
9
BINOMIAL DISTRIBUTIONMEAN 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 DISTRIBUTIONTHE 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 DISTRIBUTIONTHE PROBABILITY DENSITY
FUNCTION
18
NORMAL DISTRIBUTIONEFFECT OF CHANGING STANDARD
DEVIATION
19
NORMAL DISTRIBUTIONEFFECT OF CHANGING MEAN
SD,s10
20
STANDARD NORMAL DISTRIBUTION
21
STANDARD NORMAL DISTRIBUTION
22
STANDARD NORMAL DISTRIBUTIONRELATIONSHIP 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 z0 to some
positive z value. For example, the area from z0
to z1.3.041.34 is 0.4099.