Econ%203790:%20Business%20and%20Economics%20Statistics

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Econ 3790 Business and Economics Statistics

• Instructor Yogesh Uppal
• Email yuppal_at_ysu.edu

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Lecture Slides 5

• Random Variables
• Probability Distributions
• Discrete Distributions
• Discrete Uniform Probability Distribution
• Binomial Probability Distribution
• Continuous Distribution
• Normal Distribution

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Random Variables
A random variable is a numerical description of
the outcome of an experiment.
A discrete random variable may assume either a
finite number of values or an infinite sequence
of values.
A continuous random variable may assume any
numerical value in an interval or collection of
intervals.
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Example JSL Appliances

• Discrete random variable with a finite number of
  values

Let x number of TVs sold at the store in one
day, where x can take on 5 values (0, 1, 2, 3,
4)
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Example JSL Appliances

• Discrete random variable with an infinite
  sequence of values

Let x number of customers arriving in one
day, where x can take on the values 0, 1, 2, .
. .
We can count the customers arriving, but there
is no finite upper limit on the number that might
arrive.
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Random Variables
Type
Question
Random Variable x
Family size
x Number of dependents reported on tax
return
Discrete
Continuous
x Distance in miles from home to the
store site
Distance from home to store
Own dog or cat
Discrete
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)
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Discrete Probability Distributions
The probability distribution for a random
variable describes how probabilities are
distributed over the values of the random
variable.
We can describe a discrete probability
distribution with a table, graph, or equation.
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Discrete Probability Distributions
The probability distribution is defined by a
probability function, denoted by p(x), which
provides the probability for each value of the
random variable.
The required conditions for a discrete
probability function are
p(x) gt 0
?p(x) 1
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Discrete Probability Distributions

• Using past data on TV sales,

• a tabular representation of the probability
• distribution for TV sales was developed.

Number Units Sold of Days 0
80 1 50 2 40 3
10 4 20 200
x p(x) 0 .40 1 .25
2 .20 3 .05 4 .10
1.00
80/200
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Discrete Probability Distributions

• Graphical Representation of Probability
  Distribution

Probability
0 1 2 3 4
Values of Random Variable x (TV sales)
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Expected Value and Variance
The expected value, or mean, of a random
variable is a measure of its central
location.
The variance summarizes the variability in the
values of a random variable.
The standard deviation, ?, is defined as the
positive square root of the variance.
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Expected Value
x p(x) xp(x) 0 .40
.00 1 .25 .25 2 .20
.40 3 .05 .15 4 .10
.40 E(x) 1.20
expected number of TVs sold in a day
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Variance and Standard Deviation
(x - ?)2
p(x)
p(x)(x - ?)2
x
x - ?
-1.2 -0.2 0.8 1.8 2.8
1.44 0.04 0.64 3.24 7.84
0 1 2 3 4
.40 .25 .20 .05 .10
.576 .010 .128 .162 .784
Variance of daily sales s 2 1.660
Standard deviation of daily sales 1.2884 TVs
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Types of Discrete Probability Distributions

• Uniform
• Binomial

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Discrete Uniform Probability Distribution
The discrete uniform probability distribution is
the simplest example of a discrete probability
distribution given by a formula.
The discrete uniform probability function
is
p(x) 1/n
the values of the random variable are equally
likely
where n the number of values the random
variable may assume
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Discrete Uniform Probability Distribution

• Suppose, instead of looking at the past sales of
  the TVs, I assume (or think) that TVs sales have
  a uniform probability distribution, then the
  example done above would change as follows

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Expected Value
x p(x) xp(x) 0 .2
.00 1 .2 .20 2 .2
.40 3 .2 .60 4 .2
.80 E(x) 2.0
expected number of TVs sold in a day
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Variance and Standard Deviation
(x - ?)2
p(x)
p(x)(x - ?)2
x
x - ?
-2.0 -1.0 0.0 1.0 2.0
4.0 1.0 0.0 1.0 4.0
0 1 2 3 4
.2 .2 .2 .2 .2
0.8 0.2 0.0 0.2 0.8
Variance of daily sales s 2 2.0
Standard deviation of daily sales 1.41 TVs
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Example I am bored

• Imagine this situation. There is heavy snowstorm.
  Everything is shut down. You and everybody in
  your family have to stay home. You are utterly
  bored. You catch hold of your sibling and get him
  or her to play this game.
• The game is to bet on the toss of a coin.

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Example I am bored

• If it turns up heads exactly once in three
  tosses, you win or otherwise you lose.
• Lets call the event of getting heads on anyone
  trial as a success. Similarly, the event of
  getting tails is a failure.
• Suppose the probability of getting heads (or of a
  success) is 0.6.
• The big question is that you want to find out the
  probability of getting exactly 1 head on three
  tosses.

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Tree Diagram
Trial 2
Outcomes
Trial 1
Trial 3
H
HHH (0.6)3(0.4)0 0.216
H
HHT (0.6)2(0.4)10.144
T
H
HTH (0.6)2(0.4)1 0.144
H
T
T
HTT (0.6)1(0.4)2 0.096
H
THH (0.6)2(0.4)1 0.144
H
T
T
THT (0.6)1(0.4)2 0.096
H
T
TTH (0.6)1(0.4)2 0.096
T
TTT (0.6)0(0.4)3 0.064
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So, what is the probability of you winning the
game?

• What is the random variable here?
• What is the probability of getting 2 heads in
  three tosses?
• P(HHT) P(HTH) P(THH)
• 0.144 0.144 0.144
• 0.432
• Or 43.2
• How does the probability distribution of our
  Random variable look?

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Binomial Distribution
1. The experiment consists of a sequence of n
identical trials.
2. Two outcomes, success and failure, are
possible on each trial.
3. The probability of a success, denoted by p,
does not change from trial to trial.
4. The trials are independent.
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Binomial Distribution
Our interest is in the number of successes
occurring in the n trials.
We let x denote the number of successes
occurring in the n trials.
Binomial Distribution is highly useful when the
number of trials is large.
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Binomial Distribution

• Binomial Probability Function

where n the number of trials p the
probability of success on any one trial
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Counting Rule for Combinations

• Another useful counting rule (esp. when n is
  large) enables us to count the number of
  experimental outcomes when x objects are to be
  selected from a set of N objects.

• Number of Combinations of n Objects Taken x at a
  Time

where n! n(n - 1)(n - 2) . . . (2)(1)
x! x(x - 1)(x- 2) . . . (2)(1) 0!
1
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Example I am bored

• Using binomial distribution, the probability of 1
  head in 3 tosses is

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Example I am bored

• Suppose, you won. But knowing your sibling, she
  or he says that bet was getting exactly 2 heads
  in 3 tosses. Since you are bored, you have no
  choice but continuing to play

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Example I am bored

• She again cheats. She says that bet was getting
  at least 2 heads in 3 tosses.
• What does this mean Getting 2 or more heads P(2
  heads) P(3 heads)

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Example I am bored
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Binomial Distribution
Expected Value
Variance
Standard Deviation
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Example I am bored

• Mean (or expected value)
• Variance
• Standard Deviation

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Chapter 6 Continuous Probability Distributions

• Normal Probability Distribution

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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.

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Normal Probability Distribution

• It has been used in a wide variety of
  applications

Heights of people
Scientific measurements
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Normal Probability Distribution

• It has been used in a wide variety of
  applications

Test scores
Amounts of rainfall
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Normal Distributions

• 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 curve between x1
  and x2.

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Normal Probability Distribution

• Characteristics

The distribution is symmetric its skewness
measure is zero.
x
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Normal Probability Distribution

• Characteristics

The highest point on the normal curve is at the
mean, which is also the median and mode.
x
Mean m
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Normal Probability Distribution

• Characteristics

The entire family of normal probability
distributions is defined by its mean m and its
standard deviation s .
Standard Deviation s
x
Mean m
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Normal Probability Distribution

• Characteristics

The mean can be any numerical value negative,
zero, or positive. The following shows different
normal distributions with different means.
x
-10
0
20
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Normal Probability Distribution

• Characteristics

The standard deviation determines the width of
the curve larger values result in wider, flatter
curves.
s 15
s 25
x
Same Mean
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Normal Probability Distribution

• Characteristics

Probabilities for the normal random variable
are given by areas under the curve. The total
area under the curve is 1 (.5 to the left of the
mean and .5 to the right).
.5
.5
x
Mean m
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Standardizing the Normal Values or the z-scores

• Z-scores can be calculated as follows

• 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
A standard normal distribution is a normal
distribution with mean of 0 and variance of 1.
If x has a normal distribution with mean (µ) and
Variance (s), then z is said to have a standard
normal distribution.
s 1
z
0
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Example Air Quality

• I collected this data on the air quality of
  various cities as measured by particulate matter
  index (PMI). A PMI of less than 50 is said to
  represent good air quality.
• The data is available on the class website.
• Suppose the distribution of PMI is approximately
  normal.

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Example Air Quality

• The mean PMI is 41 and the standard deviation is
  20.5.
• Suppose I want to find out the probability that
  air quality is good or what is the probability
  that PMI is greater than 50.