Probability distributions

1
Probability distributions

• Binomial probability distribution (ASW, section
  5.4)
• Using Excel for the binomial (ASW, pp. 222-223)
• Uniform probability distribution (ASW, section
  6.1)
• Normal probability distribution (ASW, section
  6.2)
• Bring the text to class on Monday and Wednesday,
  Sept. 29 and October 1. We will be using Tables
  1 and 5 of Appendix B of ASW.

Notes for September 29, 2008
2
Variance (ASW, 195)

• The variance of a probability distribution is the
  expected value of the squares of the differences
  of the random variable x from the mean µ.
  Symbolically,
• Var(x) s2 ?(x µ)2 f(x)
• The Greek symbol s is sigma.
• The variance can be difficult to calculate and
  interpret. It is in units that are the square of
  the random variable x. Partly because of this,
  in statistical work it is more common to use the
  square root of the variance or s. The standard
  deviation has the same units as x.

3
Variance of x, number of females selected
If a random sample of 3 persons is obtained from
a large population composed of half females and
half males, the expected number of females
selected is µ 1.5. The variance of the number
of females selected is Var(x) s2 ?(x µ)2
f(x) 0.75. The standard deviation is the
square root of 0.75, so that s 0.866.
4
Sample and population variance

• The variance of a sampling distribution is (ASW,
  195)
• This is equivalent to the variance of a
  population (ASW, 92)
• Note that the variance of a sample is

Var(x) s2 ?(x µ)2 f(x)
5
Unbiased estimator

• The expected value of s2 is equal to s2, a
  characteristic that is referred to as an unbiased
  estimate. That is,
• Using (n-1) in the denominator of s2, rather
  than n, produces this unbiased estimate.
• The concept of biased and unbiased estimators is
  important in constructing good estimators and is
  a major consideration in econometric work.
• When using Excel to estimate mean and standard
  deviation, make sure you use the proper formulae.

6
Binomial probability distribution (ASW, 200)

• A binomial experiment is a probability experiment
  with the following characteristics
• The experiment has n identical trials.
• Two outcomes are possible on each trial one
  trial is termed a success and the other is termed
  a failure.
• The probability of a success occurring on each
  trial is p. This probability p is the same on
  each trial.
• Since the outcome must either be a success or
  failure, a failure is the complement of a success
  and the probability of a failure is 1-p. (Some
  texts refer to this probability as q, that is,
  q1-p).
• The trials are independent of each other.

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Given the above conditions

• The binomial probability distribution provides
  the probability of x successes in n trials, where
  x0, 1 ,2, 3, , n.
• Note that there are only two parameters that
  determine binomial probabilities
• n the number of trials.
• p the probability of success.
• Successive trials must be independent of each
  other. That is, the outcome of any one trial
  must not affect the probability of success or
  failure for any other trial.
• P (success ? failure on any other trial) p
• P (success ? success on any other trial) p

8
i
Example number of females selected in a random
sample of size 3 from a large population of half
males and half females.
x is the number of females selected and f(x) is
the probability of x females being selected
The above distribution is a binomial probability
distribution with success defined as selecting a
female. There are n 3 independent trials, the
probability of success is p 0.5, and x is the
number of successes. In this experiment,
selecting a male is termed a failure, and the
probability of selecting a male is 1-p 1-0.5
0.5.
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Formula for binomial probability
If n is the number of trials of the binomial
experiment and p is the probability of success,
then the probability of x successes in n trials
of the experiment is given by the probability
function f(x), defined as follows
10
Using the binomial formula
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Combinations and permutations (ASW, 146-147)

• Permutations the number of ways of arranging N
  objects, taken n at a time, where the order of
  the objects is taken into account, is
• Where is the number of possible
  combinations of N objects, taken n at a time,
  where the order of the objects does not matter.
  

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Rationale for the binomial formula

• Probability of x successes and (n-x) failures is
• This is and represents the
  probability of any particular sequence of x
  successes and (n-x) failures.
• And there are ways of arranging these x
  successes and (n-x) failures. To obtain the
  probability of x successes in n trials, multiply
  the probability of any particular sequence by
  this combination.

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Example selection of Saskatchewan workers,
classified by years of education and wages and
salaries

• From all these workers, randomly select 13
  workers with 14-17 years of education. What is
  the probability that exactly 8 of these will have
  incomes of 45,000 or more? Probability of 8 or
  more?
• A random sample from a large population means
  that successive selections are independent of
  each other. There are n 13 workers selected.
  If success is defined as the probability of
  selecting a worker with an income of 45,000 or
  more, the probability of success p 82/230
  0.357.
• Probability of 8 with 45,000 or more income
  0.0373. See the following slides for the
  calculation.

14
Using the formula
15
Probabilities to 3 decimal places
The probability of 8 or more successes is the sum
of the probabilities of 8, 9, 10, 11, 12, or 13
successes. This is 0.0373 0.0115 0.0026
0.0004 0.0000 0.0000 0.0518.
16
Using an Excel worksheet to obtain the
probabilities
17
Formula in Excel

• n13 is in cell a1 and p0.357 is in cell a2.

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Mean and standard deviation

• For a binomial distribution with n trials and p
  as the probability of success, the mean or
  expected value and variance of the random
  variable x is
• For the sex distribution of n 3 individuals,
  the expected number of females selected is 3
  0.5 1.5 and the variance is 3 0.5 0.5
  0.75, as we previously determined.
• For the experiment of selecting 13 individuals,
  the mean number of those with 14-17 years of
  education is 13 0.357 4.64, the variance is
  13 0.357 0.643 2.984, and the standard
  deviation is 1.727.

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Examples where binomial could be applied

• The probability of ten or more heads when
  flipping a coin twelve times.
• The probability of 6 threes in 15 rolls of a die.
• The probability of selecting 56 or more
  unemployed persons in a random sample of 500
  workers in the province of Saskatchewan.
• The probability that the tax form has been
  correctly completed in a random sample of 500
  Canadian taxpayers.
• The probability that more than 1/3 of a sample
  1,000 Saskatchewan residents has a university
  degree.

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Why might the binomial not apply in the following?

• The probability that there will be snow on 20 or
  more days in January?
• The probability of 6 threes and 7 fives in 25
  rolls of a die.
• The probability that the UR Rams win all of their
  remaining football games?
• The probability that the Conservatives win 155 or
  more seats, among the 308 up for election, in the
  coming federal election.
• The probability that 10 or more automobiles in a
  car dealers lot in Regina will have defective
  transmissions.
• The probability that fifty or more clients of the
  Regina Food Bank , during the month of October,
  will be unemployed.

21
Extending the binomial

• When the number of trials of a binomial
  experiment is large, ie. if n is large, then it
  is time-consuming to compute binomial
  probabilities without a computer.
• In this case, it is possible to use the normal
  distribution to approximate the binomial
  probabilities. See ASW, section 6.3.
• In addition, we may not be as interested in the
  number of successes as in the proportion of
  successes. In this case, the normal
  approximation can be used to obtain probabilities
  for the proportion p of the times that a success
  occurs. See ASW, section 7.6.

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Later on Monday or on Wednesday

• Uniform probability distribution.
• Normal probability distribution.
• Normal approximation to the binomial probability
  distribution.