MGMT 242

1
Random Variables and Probability Distributions
Chapter 4

• Never draw to an inside straight.
• from Maxims Learned at My Mothers Knee

2
Goals for Chapter 4

• Define Random Variable, Probability Distribution
• Understand and Calculate Expected Value
• Understand and Calculate Variance, Standard
  Deviation
• Two Random Variables--Understand
• Statistical Independence of Two Random Variables
• Covariance Correlation of Two Random Variables
• Applications of the Above

3
Random Variables

• Refers to possible numerical values that will be
  the outcome of an experiment or trial and that
  will vary randomly
• Notation uppercase letters for last part of
  alphabet--X, Y, Z
• Derived from hypothetical population (infinite
  number), the members of which will have different
  values of the random variable
• Example--let Y height of females between 18 and
  20 years of age population is infinite number
  of females between 18 and 20
• Actual measurements are carried out on a sample,
  which is randomly chosen from population.

4
Probability Distributions

• A probability distribution gives the probability
  for a specific value of the random variable
• P(Y y) gives the probability distribution when
  values for P are specified for specific values of
  y
• example tossed a coin twice(fair coin) Y is
  the number of heads that are tossed possible
  events TT, TH, HT, HH each event is equally
  probable if coin is a fair coin, so probability
  of Y0 (event TT) is 1/4 probability of Y 2
  is 1/4 probability of Y 1 is 1/4 1/4 1/2
  
• Thus, for example
• P(Y0) 1/4
• P(Y1) 1/2
• P(Y2) 1/4

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Probability Distributions--Discrete Variables

• Notation P(Yy) or, occasionally, PY(y) (with
  y being some specific value
• PY(y) is some value when Yy and 0 otherwise
• Example--(Ex. 4.1) 8 people in a business group,
  5 men, 3 women. Two people sent out on a
  recruiting trip. If people randomly chosen,
  find PY(y) for Y being the number of women sent
  out on the recruiting trip.
• Solution (see board work)
  PY(2) 3/28 PY(1)
  15/28 PY(0) 10 /28

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Cumulative Probability Distribution--Discrete
Variables

• Notation P(Y? y) means the probability that Y is
  less than or equal to y FY(y) is the same.
• Complement rule P(Y gt y) 1 - FY(y).
• Example (previous scenario, 5 men, 3 women,
  interview trips). What is the probability that
  at least one woman will be sent on an interview
  trip? (Note at least means 1 or greater than
  1)
• P (Ygt 0) 1 - FY(0) 1 - PY(0) 1 - 10/28
  18/28
• In this example, its just as easy to calculate
  P(Y? 1) P(Y1)
  P(Y2) 15 /28 3 / 28, but many times its not
  as easy.

7
Expectation Values, Mean Values

• The expectation value of a discrete random
  variable Y is defined by the relation E(Y)
  ?iPY( yi) yi , that is to say, the sum of all
  possible values of Y, with each value weighted by
  the probability of the value.
• Note that E(Y) is the mean value of Y E(Y) is
  also denoted as lt Y gt .
• Example (for previous case, 8 people, 5 men, 3
  women,) If Y is the number of women on an
  interview trip (Y 0, 1, or 2), then
  E(Y) (3/28) x 2 (15/28) x 1
  (10/28)x0 21/28 3/4

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Variance of a Discrete Random Variable

• The variance of a discrete random variable, V(Y),
  is the expectation of the square of the
  deviation from the mean (expected value) V(Y) is
  also denoted as Var(Y)
• V(Y) E(Y- E(Y))2 ?iPY( yi) yi - E(Y)2
• V(Y) E(Y2) - (E(Y))2
• The second formula for V(Y) is derived as
  follows
• V(Y) ?iPY( yi) yi2 - 2 yi E(y) (E(Y))2
• or V(Y) E(Y2) - 2 E(y) E(y) (E(Y))2 E(Y2)
  - (E(Y))2
• Example (previous, 5 men, 3 women, etc..)
• V(Y) (3/28)(2- 3/4)2 (15/28) (1- 3/4)2
  (10/28) (0- 3/4)2, or V(Y) (3/28) 22 (15/28)
  12 0 - (3/4)2 27/28

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Expectation Values--Another ExampleExercise
4.25, Investment in 2 Apartment Houses
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Continuous Random Variables

• Reasons for using a continuous variable rather
  than discrete
• Many, many values (e.g. salaries)--too many to
  take as discrete
• Model for probability distribution makes it
  convenient or necessary to use a continuous
  variable--
• Uniform Distribution (any value between set
  limits equally likely)
• Exponential Distribution (waiting times, delay
  times)
• Normal (Gaussian) Distribution, the Bell Shaped
  Curve (many measurement values follow a normal
  distribution either directly or after an
  appropriate transformation of variables also
  mean values of samples follow a normal
  distribution, generally.)

11
Probability Density and Cumulative Density
Functions for Continuous Variables

• Probability density function, fX(x) defined
• P(x? X ? xdx) fX(x) dx, that is, the
  probability that the random variable X is between
  x and xdx is given by fX(x) dx
• Cumulative density function, FX(x), defined
• P(X ? x ) FX(x)
• FX(x) ? fX(x)dx, where the integral is taken
  from the lowest possible value of the random
  variable X to the value x.

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Probability Density and Cumulative Density
Functions for Continuous Variables, Example
Exercise 4.12

• Model for time, t, between successive job
  applications to a large computer is given by
  FT(t) 1 - exp(-0.5t).
• Note that FT(t) 0 for t 0 and that FT(t)
  approaches 1 for t approaching infinity.
• Also, fT(t) the derivative of FT(t), or
  fT(t) 0.5exp(-0.5t)

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Probability Density and Cumulative Density
Functions for Continuous Variables--
Example, Problem 4.12
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Expectation Values for Continuous Variables

• The expectation value for a continuous variable
  is taken by weighting the quantity by the
  probability density function, fY(y), and then
  integrating over the range of the random variable
• E(Y) ? y fY(y) dy
• E(Y), the mean value of Y, is also denoted as ?Y
• E(Y2) ? y fY(y) dy
• The variance is given by V(Y) E(Y2) - (?Y)2

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Continuous Variables--Example

• Ex. 4.18, text. An investment company is going
  to sell excess time on its computer it has
  determined that a good model for the its own
  computer usage is given by the probability
  density function
  fY(y) 0.000937540-0.1(y-100)2
  for 80 lt y lt 120 fY(y) 0, otherwise.
• The important things to note about this
  distribution function can be determined by
  inspection
• there is a maximum in fY(y) at y 100
• fY(y) is 0 at y80 and y 120
• fY(y) is symmetric about y100 (therefore E(y)
  100 and FY(y)1/2 at y 100).
• fY(y) is a curve that looks like a symmetric hump.

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Two Random Variables

• The situation with two random variables, X and Y,
  is important because the analysis will often show
  if there is a relation between the two, for
  example, between height and weight years of
  education and income blood alcohol level and
  reaction time.
• We will be concerned primarily with the
  quantities that show how strong (or weak) the
  relation is between X and Y
• The covariance of X and Y is defined by
  Cov(X,Y) E(X-?X)(Y- ?Y) E(XY) -?X?Y
• The correlation of X and Y is defined by
  Cor(X,Y) Cov(X,Y) / ?(V(X)V(Y)
  Cov(X,Y)/(?X ?Y)

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Properties of Covariance, Correlation

• Cov(X,Y) is positive (gt0) if X and Y both
  increase or both decrease together Examples
• height and weight years of education and
  salary
• Cov(X,Y) is zero if X and Y are statistically
  independent
  Cov(X,Y) E(XY) -?X?Y
  E(X)E(Y)-?X?Y ?X?Y - ?X?Y 0 Examples adult
  hat size and IQ
• Cov(X,Y) is negative if Y increases while X
  decreases Example
• annual income, number of bowling games per year.
• (no disrespect meant to bowlers).

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Statistical Independence of Two Random Variables

• If two random variables, X and Y, are
  statistically independent, then
• P(Xx, Yy) P(Xx) P(Yy),
• that is to say, the joint probability density
  function can be written as the product of
  probability density functions for X and Y.
• Cov(X, Y) 0
• This follows from the above relation
• Cov(X,Y) E(X-µX) (Y-µY) E(X-µX)E(Y-µY)
• (µX-µX) (µY-µY) 0