QMS 6351 Statistics and Research Methods Probability and Probability distributions Chapter 4, page 161 Chapter 5 (5.1) Chapter 6 (6.2)

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QMS 6351Statistics and Research
MethodsProbability and Probability
distributionsChapter 4, page 161Chapter 5
(5.1)Chapter 6 (6.2)

• Prof. Vera Adamchik

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Probability

• Probability is a numerical measure of the
  likelihood that a specific event will occur.
• Properties of Probability
• The probability of an event always lies in the
  range zero to 1, including zero (an impossible
  event) and 1 (a sure event)
• The sum of all probabilities for an experiment is
  always 1.

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Assigning Probabilities to Experimental Outcomes

• Classical Method
• Assigning probabilities based on the assumption
  of equally likely outcomes.
• Relative Frequency Method
• Assigning probabilities based on experimentation
  or historical data.
• Subjective Method
• Assigning probabilities based on the assignors
  judgment.

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Classical Method

• If an experiment has n possible outcomes,
  this method would assign a probability of 1/n to
  each outcome.
• Example
• Experiment Rolling a die
• Sample Space S 1, 2, 3, 4, 5, 6
• Probabilities Each sample point has a 1/6
  chance of occurring.

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Relative Frequency as an Approximation of
Probability

• If an experiment is repeated n times and an event
  A is observed f times, then, according to the
  relative frequency concept of probability

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Law of Large Numbers

• Relative frequencies are not probabilities but
  approximate probabilities. However, if the
  experiment is repeated a very large number of
  times, the approximate probability of an outcome
  obtained from the relative frequency will
  approach the actual probability of that outcome.
  This is called the Law of Large Numbers.

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Subjective Method

• Subjective probability is the probability
  assigned to an event based on subjective
  judgement, experience, information, and belief.
• The best probability estimates often are obtained
  by combining the estimates from the classical or
  relative frequency approach with the subjective
  estimates.

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Objective and Subjective Probability

• When probabilities are assessed in ways that
  are consistent with the classical or relative
  frequency determination of probability, we call
  them objective probabilities. Objective and
  subjective probabilities are fundamentally
  different.

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• If a number of people assign the probability of
  an event objectively, each individual will arrive
  at the same answer, provided they did the
  calculations properly.
• If a number of people assign the probability of
  an event subjectively, each individual will
  arrive at his or her own answer.
• As a consequence, not all probability theory and
  methods that can be applied to objective
  probabilities can be applied to subjective ones.

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Random variables

• A random variable is a numerical description of
  the outcome of an experiment.

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Discrete random variable

• A random variable is discrete if the set of
  outcomes is either finite in number (e.g., tail
  head 1,2,3,4,5,6 face value of a die) or
  countably infinite (e.g., the number of children
  in a family 0,1,2,3,).

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Continuous random variable

• A random variable is continuous if the set of
  outcomes is infinitely divisible and, hence, not
  countable. A continuous random variable may
  assume any numerical value in an interval. For
  example, temperature may assume any value between
  42F and 56F, or between 42F and 45F, or between
  42F and 43F, or between 42F and 42,5F etc.

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Question Random Variable x Type
Family size x Number of dependents reported on tax return Discrete
Distance from home to store x Distance in miles from home to the store site Continuous
Own dog or cat x 1 if own no pet 2 if own dog(s) only 3 if own cat(s) only 4 if own dog(s) and cat(s) Discrete
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Discrete probability function

• The probability function (denoted by f(x)) for a
  discrete random variable lists all the possible
  values that the random variable can assume and
  their corresponding probabilities. For example,
  f(head) 0.5, f(tail) 0.5.
• We can describe a discrete probability
  distribution with a table, graph, or equation.

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Continuous probability function

• With continuous random variables, the counterpart
  of the probability function is the probability
  density function, also denoted by f(x).
• However, important difference exists between
  probability distributions for discrete and
  continuous variables

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Difference (1)

• in the continuous case, f(x) is a counterpart of
  probability function f(x), but is called
  probability density function, p.d.f

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Difference (2)

• p.d.f. provides the value of the function at any
  particular value of x it does not directly
  provide the probability of the random variable
  assuming some specific value

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Difference (3)

• Probability is represented by the area under the
  graph. Because the area under the curve (line)
  above any single point is 0, P(x value) 0.
• In the continuous case,
• P(a lt x lt b)
• P(a lt x lt b) P(a lt x lt b)
• P(a lt x lt b).

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The probability of the random variable
assuming a value within some given interval from
x1 to x2 is defined to be the area under the
graph of the probability density function between
x1 and x2.
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Continuous Probability Distributions

• It is not possible to talk about the probability
  of the random variable assuming a particular
  value.
• Instead, we talk about the probability of the
  random variable assuming a value within a given
  interval.
• The probability of the random variable assuming a
  value within some given interval from x1 to x2 is
  defined to be the area under the graph of the
  probability density function between x1 and x2.

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The Normal Probability Distribution

• The normal probability distribution is the most
  important distribution for describing a
  continuous random variable.
• It is widely used in statistical inference.
• It has been used in a wide variety of
  applications including
• Heights of people
• Test scores
• Rainfall amounts

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The Normal Probability Distribution

• The Normal Curve
• The shape of the normal curve is often
  illustrated as a bell-shaped curve.
• The highest point on the normal curve is at the
  mean, which is also the median and mode of the
  distribution.
• The normal curve is symmetric about the mean.
• The tails of a normal distribution curve extend
  indefinitely in both directions without touching
  or crossing the horizontal axis.

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The Normal Probability Distribution

• Graph of the Normal Probability Density Function

f (x )
x
?
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The Normal Probability Distribution

• The entire family of normal probability
  distributions is defined by its mean µ and its
  standard deviation s .

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The Normal Probability Distribution

• Normal Probability Density Function
• where
• ? mean
• ? standard deviation
• ? 3.14159
• e 2.71828

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The mean determines the position on the curve
with respect to other normal curves. The mean can
be any numerical value negative, zero, or
positive.
x
-10
0
25
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The standard deviation determines the width of
the curve larger values result in wider, flatter
curves.
s 15
s 25
x
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Probabilities for the normal random variable are
given by areas under the curve. The total area
under the curve is 1 (0.5 to the left of the mean
and 0.5 to the right).
.5
.5
x
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Standard Normal Probability Distribution

• A random variable that has a normal distribution
  with a mean of zero and a standard deviation of
  one is said to have a standard normal probability
  distribution.
• The letter z is commonly used to designate this
  normal random variable.

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Converting to the Standard Normal Distribution

• For a normal variable x, a particular value can
  be converted to a z value
• We can think of z as a measure of the number of
  standard deviations x is from ?.

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Standard Normal Probability Distribution
s 1
z
0
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Example 1 P/E Ratio

• The price-earnings (P/E) ratio for a company
  is an indication of whether the stock of that
  company is undervalued (P/E is low) or overvalued
  (P/E is high). Suppose the P/E ratios of all
  companies have a normal distribution with a mean
  15 and a standard deviation of 6.
• If a P/E ratio of more than 20 is considered
  to be a relatively high ratio, what percentage of
  all companies have high P/E ratios?
• P(x gt 20) ?

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Example 1 Solution steps

• Step 1 Convert x to the standard normal
  distribution.
• z (x - µ)/s
• (20 - 15)/6
• 0.83
• Step 2 Find the area under the standard normal
  curve to the left of z 0.83.

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Cumulative probability table for the standard
normal distribution
P(z lt .83)
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• Step 3 Compute the area under the standard
  normal curve to the right of z 0.83.

P(z gt .83) 1 P(z lt .83) 1-
.7967 .2033
P(x gt 20)
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Area 1 - .7967 .2033
Area .7967
z
0
.83
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Example 2 GMAT score

• Most business schools require that every
  applicant for admission to a degree program take
  the GMAT. Suppose the GMAT scores of all students
  have a normal distribution with a mean of 50 and
  a standard deviation of 90. What should your
  score be so that only 5 of all the examinees
  score higher than you do?
• x0.05 ?

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Area .9500
Area .0500
z
0
z.05
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Example 2 Solution steps

• Step 1 Find the z-value that cuts off an area of
  .05 in the right tail of the standard normal
  distribution.

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We look up the complement of the tail area (1 -
.05 .95)
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• Step 2 Convert z0.05 to the corresponding value
  of x
• x ? z.05?
• ??? 550 1.64590
• 698.05
• Your score should be 698 so that only 5 of
  all the examinees score higher than you do.