Lecture 14: Probability Theory and Random Variables

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Lecture 14 Probability Theory and Random
Variables

• Professor Aurobindo Ghosh
• E-mail ghosh_at_galton.econ.uiuc.edu

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Probability Rules

• Complement rule
• Each simple event must belong to either A or
  .Since the sum of the probabilities assigned to
  a simple event is one, we have for any event A

P(A) 1 - P(A)
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• Addition rule
• For any two events A and B

P(A or B) P(A) P(B) - P(A and B)
P(A) 6/13

P(B) 5/13
_
A
P(A and B) 3/13
P(A or B) 8/13
B
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• Multiplication rule
• For any two events A and B
• When A and B are independent

P(A and B) P(A)P(BA) P(B)P(AB)
P(A and B) P(A)P(B)
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• Example 6.3

• A stock market analyst feels that
• the probability that a certain mutual fund will
  receive increased contributions from investors is
  0.6.
• the probability of receiving increased
  contributions from investors becomes 0.9 if the
  stock market goes up.
• the probability of receiving increased
  contributions from investors drops below 0.6 if
  the stock market drops.
• there is a probability of 0.5 that the stock
  market rises.
• The events of interest are
• A The stock market rises
• B The company receives increased contribution.

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• Calculate the following probabilities
• The probability that both A and B will occur is
  P(A and B). Sharp increase in earnings.
• The probability that either A or B will occur
  isP(A or B). At least moderate increase in
  earning.
• Solution
• P(A) 0.5 P(B) 0.6 P(BA) 0.9
• P(A and B) P(A)P(BA) (.5)(.9) 0.45
• P(A or B) P(A) P(B) - P(A and B) .5 .6 -
  .45 0.65

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Random Variables and Probability Distributions

• A random experiment is a function that assigns a
  numerical value to each simple event in a sample
  space.
• A random variable reflects the aspect of a random
  experiment that is of interest to us.
• There are two types of random variables
• Discrete random variable
• Continuous random variable.

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Discrete and Continuous Random Variables
A random variable is discrete if it can assume
only a countable number of values. A random
variable is continuous if it can assume an
uncountable number of values.
Discrete random variable
Continuous random variable
After the first value is defined the second
value, and any value thereafter are known.
After the first value is defined, any number can
be the next one
0
1
1/2
1/4
1/16
Therefore, the number of values is countable
Therefore, the number of values is uncountable
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Discrete Probability Distribution

• A table, formula, or graph that lists all
  possible values a discrete random variable can
  assume, together with associated probabilities,
  is called a discrete probability distribution..
• To calculate P(X x), the probability that the
  random variable X assumes the value x, add the
  probabilities of all the simple events for which
  X is equal to x.

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• Example
• Find the probability distribution of the random
  variable describing the number of heads that
  turn-up when a coin is flipped twice.
• Solution

Simple event x Probability HH 2 1/4 H
T 1 1/4 TH 1 1/4 TT 0 1/4
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• Requirements of discrete probability distribution
• If a random variable can take values xi, then the
  following must be true

• The probability distribution can be used to
  calculate probabilities of different events.
• Example continued

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• Probabilities as relative frequencies
• In practice, often probabilities are estimated
  from relative frequencies
• Example
• The number of cars a dealer is selling daily were
  recorded in the last 100 days. This data was
  summarized as follows

Daily sales Frequency 0 5 1 15 2 35 3
25 4 20 100

• Estimate the probability distribution.
• State the probability of selling more than 2
  cars a day.

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• Solution
• From the table of frequencies we can calculate
  the relative frequencies, which becomes our
  estimated probability distribution

Daily sales Relative Frequency 0 5/100.05 1
15/100.15 2 35/100.35 3 25/100.25 4 20/
100.20 1.00
X
0 1 2 3 4

• The probability of selling more than 2 a day is

P(Xgt2) P(X3) P(X4) .25 .20
.45
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6.5 Expected Value and Variance

• The expected value
• Given a discrete random variable X with values
  xi, that occur with probabilities p(xi), the
  expected value of X is

• The expected value of a random variable X is the
  weighted average of the possible values it can
  assume, where the weights are the corresponding
  probabilities of each xi.

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Laws of Expected Value

• E(c) c
• E(cX) cE(X)
• E(X Y) E(X) E(Y)E(X - Y) E(X) - E(Y)
• E(XY) E(X)E(Y) if X and Y are independent
  random variables.

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Variance

• Let X be a discrete random variable with
  possible values xi that occur with
  probabilities p(xi), and let E(xi) m. The
  variance of X is defined to be

• The variance is the weighted average of the
  squared deviations of the values of X from their
  mean m, where the weights are the corresponding
  probabilities of each xi.

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• Standard deviation
• The standard deviation of a random variable X,
  denoted s, is the positive square root of the
  variance of X.
• Example 6.5
• The total number of cars to be sold next week is
  described by the following probability
  distribution
• Determine the expected value and standard
  deviation of X, the number of cars sold.

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