Lecture 15: Applications of Discrete Probability Distributions

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Lecture 15 Applications of Discrete Probability
Distributions

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

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Bivariate Distributions

• To consider the relationship between two random
  variables, the bivariate (or joint) distribution
  is needed.
• Bivariate probability distribution
• The probability that X assumes the value x, and Y
  assumes the value y is denoted
• p(x,y) P(Xx, Y y)

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• Example 6.7
• Xavier and Yvette are two real estate agents.
  Let X and Y denote the number of houses that
  Xavier and Yvette will sell next week,
  respectively.
• The bivariate (joint) probability distribution

p(0,0)
P(Y1), the marginal probability.
p(0,1)
p(0,2)
P(X0) The marginal probability
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0.42
x p(x) y p(y) 0
.4 0 .6 1 .5
1 .3 2 .1 2
.1 E(X) .7 E(Y) .5 V(X) .41
V(Y) .45
p(x,y)
0.21
0.12
0.06
X
y0
0.06
0.03
0.07
0.02
y1
0.01
Y
y2
X0
X2
X1
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Calculating Conditional Probability

• Example 6.7 - continued

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Conditions for independence

• Two random variables are said to be independent
  when
• This leads to the following relationship for
  independent variables
• Example 6.7 - continued
• Since P(X0Y1).7 but P(X0).4, The variables
  X and Y are not independent.

P(XxYy)P(Xx) or P(YyXx)P(Yy).
P(Xx and Yy) P(Xx)P(Yy)
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The sum of two variables

• To calculate the probability distribution for a
  sum of two variables X and Y observe the example
  below.
• Example 6.7 - continued
• Find the probability distribution of the total
  number of houses sold per week by Xavier and
  Yvette.
• Solution
• XY is the total number of houses sold. XY can
  have the values 0, 1, 2, 3, 4.
• We find the distribution of XY as demonstrated
  next.

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The probabilities P(XY)3 and P(XY) 4 are
calculated the same way. The distribution
follows
P(XY0) P(X0 and Y0) .12
P(XY1) P(X0 and Y1) P(X1 and Y0) .21
.42 .63
P(XY2) P(X0 and Y2) P(X1 and Y1) P(X2
and Y0) .07 .06 .06 .19
.. ..
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• Expected value and variance of XY
• When the distribution of XY is known (see the
  previous example) we can calculate E(XY) and
  V(XY) directly using their definitions.
• An alternative is to use the relationships
• E(aXbY) aE(X) bE(Y)
• V(aXbY) a2V(X) b2V(Y) if X and Y are
  independent.
• When X and Y are not independent, (see the
  previous example) we need to incorporate the
  covariance in the calculations of the variance
  V(aXbY).

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Covariance

• The covariance is a measure of the degree to
  which two random variables tend to move together.

COV(X,Y) S(X-mx)(y-my)p(x,y) E(X - mx)(Y -
my) E(XY) - mxmy
Over all x,y
The expected values
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• Example 6.7 - continued
• Find the covariance of the sales variables X and
  Y, then calculate the coefficient of correlation.

• Solution
• Calculation of the expected values mx
  Sxip(xi) 0(.4)1(.5)2(.1).7my Syip(yi)
  0(.6)1(.3)2(.1).5

• Calculation of the covarianceCOV(X,Y) S(x -
  mx)(y - my)p(x,,y)
  (0-.7)(0-.5)(.12)(0-.7)(1-.5)(.21)
  (0-.7)(2-5)(.07)
  (2-..7)(2-.5)(.01) -.15

There is a negative relationship between the two
variables
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• To find how strong the relationship between X and
  Y is we need to calculate the coefficient of
  correlation.

• Calculation of the standard deviations of X and
  YV(X) S(xi-mx)2p(xi) (0-.7)2(.4)(1-.7)2(.5)
  (2-.7)2(.1).41sx V(X)1/2 .64
• In a similar manner we have V(Y) .45sy
  .451/2.67
• Calculation of r

There is a relatively weak negative relationship
between X and Y .
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• The variance of the sum of two variables X and Y
  can now be calculated using
• V(aX bY) a2V(X) b2V(Y) 2abCOV(X,Y)
  a2V(X) b2V(Y) 2abr

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• Example 6.8
• Investment portfolio diversification
• An investor has decided to invest equal amounts
  of money in two investments.
• Find the expected return on the portfolio
• If r 1, .5, 0 find the standard deviation of
  the portfolio.

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• Solution
• The return on the portfolio can be represented by
• Rp w1R1 w2R2 .5R1 .5R2

The relative weights are proportional to the
amounts invested.

• Thus, E(Rp) w1E(R1) w2E(R2)
• .5(.15) .5(.27) .21
• The variance of the portfolio return is
• V(Rp) w12V(R1) w22V(R1) 2w1w2rs1s2

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• Substituting the required coefficient of
  correlationwe have
• For r 1 V(Rp) .1056
  .3250
• For r .5 V(Rp) .0806
  .2839
• For r 0 V(Rp) .0556
  .2358

Larger diversification is expressed by smaller
correlation. As the correlation coefficient
decreases, the standard deviation decreases too.
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The Binomial Distribution

• The binomial experiment can result in only one
  out of two outcomes.
• Typical cases where the binomial experiment
  applies
• A coin flipped results in heads or tails
• An election candidate wins or loses
• An employee is male or female
• A car uses 87octane gasoline, or another gasoline.

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Binomial experiment

• There are n trials (n is finite and fixed).
• Each trial can result in a success or a failure.
• The probability p of success is the same for all
  the trials.
• All the trials of the experiment are independent.
• Binomial Random Variable
• The binomial random variable counts the number of
  successes in n trials of the binomial experiment.
• By definition, this is a discrete random variable.

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Calculating the Binomial Probability
In general, The binomial probability is
calculated by