Lecture 7: Discrete Random Variables and their Distributions

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Lecture 7 Discrete Random Variables and their
Distributions

• Devore, Ch. 3.1 - 3.3

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Topics

• Random variables
• Definition
• Types of R.V.s
• Probability distributions for discrete random
  variables
• Probability mass function
• Cumulative distribution (mass) function
• Expected values of discrete random variables
• Expected values (r.v., function, rules)
• Variance (r.v., rules)

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I. Random variables

• Definition A random variable (rv) is a rule that
  associates a value with each of the possible
  outcomes of an event
• Example if X is a rv that defines the number of
  heads that we may obtain when we flip 3 coins,
  then X can take 4 values 0,1,2,3

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Bernoulli random variables

• A random variable (rv) whose values are only 0 or
  1 is called a Bernoulli random variable
• Common Notation 0 Fail (F) 1 Success (S)
• Example Suppose X is a rv that indicates whether
  a user can login to a computer system, either all
  ports are busy (Fail) or there is at least one
  port free (Success), then X can take 2 values, 0
  or 1
• X (S)1, X(F)0

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Discrete and Continuous R.V.s

• A discrete R.V. has whole unit values either in a
  finite set or listed in an ascending infinite
  sequence (first element, second element, and so
  on)
• Example X is discrete R.V. defined to be the
  of pumps in use at a gas station
• A continuous R.V. takes any value in a given
  interval. It has an infinite (or nearly infinite)
  set of possible values in that interval.
• Example Z is a continuous R.V., defined as the
  speed at which a professional tennis player hits
  his/her first serve

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Exercise 1

• Let X the number of nonzero digits in a
  randomly selected zip code. What are the possible
  values of X ?
• Give three possible outcomes and their
  associated X values

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Different R.V.s from the Same Sample Space

• Gas Station Example
• Two Stations 1 6 pumps 2 4 pumps
• Possible rvs to study
• T Total of pumps in use
• X difference between the used at stations 1
  and 2
• U maximum of pumps in use at either station
• Z of stations having exactly 2 pumps in use

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Probability Distributions

• We describe how the total probability of a RV is
  allocated among its possible values using the
• Probability mass function, PMF, for discrete
  RVs, or
• Probability density function, PDF, for continuous
  RVs.

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II. Probability distributions for discrete random
variables

• The PMF, p(x), of a discrete R.V. is defined for
  every number x by

Re-stated for every possible value x of the
random variable, the PMF specifies the
probability of observing that value.
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Exercise 2 PMF

• The sample space of a random experiment is
  a,b,c,d,e,f, and each outcome is equally
  likely. A rv X is defined as follows
• Determine the pmf of X

p(0)
p( )
p( )
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Exercise 3

• Consider a group of five potential blood donors
  A,B,C,D,E. Only A and B have type O.
• Suppose you take five blood samples, one from
  each individual, in random order. If RV Y of
  samples necessary to identify an O individual,
  define the PMF of Y.

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Family of Probability Distributions

• When p(x) depends on a parameter, the collection
  of all probability distributions for different
  values of the parameter is called a family of
  probability distributions.

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Probability Family Example

• The probability mass function of any Bernoulli
  random variable can be expressed in terms of a
  single parameter

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Cumulative Distribution Function

• The cumulative distribution function (CDF) F(x)
  of a discrete random variable X is defined by
• Example Determine the CDF for Y if its
  probability mass function is given by

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Exercise 4

• We may derive the PMF from the CDF
• Suppose that X is the number of sick days taken
  by an employee. If the maximum number is 14,
  possible values range from 0 to 14. If

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III. Expected values of discrete random variables

• The expected value or mean value of a discrete
  random variable X can be defined as
• Where X can take values from the set D

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Exercise 5

• Let X be a discrete RV that equals the number of
  bits in error during a digital transmission.
  Suppose X can take values from 0 to 4. The PMF is
  defined as follows

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Expected Value of a Function of a Random Variable

• Let X be a discrete random variable with a set D
  of possible values. We can compute the expected
  value of any function h(X) by computing

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Exercise 6

• Let X the outcome when a fair die is rolled
  once.
• A gambler offers you the following 2 different
  options
• Do not roll the die and be paid nothing, or
• Roll the die and be paid h(X)(1/X - 1/3.5) ,
  where X is the outcome of rolling the die once.
• Should you roll the die?

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Rules of Expected Value

• E(X) is a Linear Operator, i.e.

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Exercise 7

• Use the rules for expected values to calculate

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Variance of x

• The variance of a discrete RV X, V(X), is

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Rules of Variance

• Two important rules of Variance
• Variance of a function h(X)

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Exercise 8

• A computer store purchases 3 computers at 500 a
  piece. The computers will be sold at a price of
  1000. The manufacturer will re-purchase any
  unsold computers from the computer store after a
  time period for 200 a piece. Let X be the of
  computers sold at the end of the period.