Random Variables and Distributions

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Random Variables and Distributions

• Lecture 5 Stat 700

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Basics of Random Variables and Probability
Distributions

• Consider again the experiment of tossing three
  fair coins simultaneously. The sample space and
  corresponding probabilities are given by
• S HHH, HHT, HTH, THH, HTT, THT, TTH, TTT
• Probabilities 1/8, 1/8, 1/8, 1/8, 1/8, 1/8,
  1/8, 1/8
• In a practical setting, H may represent
  success of a medical operation T represents
  failure.
• In this situation we may not really be interested
  in the elementary outcomes of S, but rather our
  interest might be on the total number of H that
  occurred in the outcome.

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Notion of a Random Variable

• When interest is on some numerical characteristic
  of the outcomes of the experiment, then we are
  led into consideration of random variables.
• Definition A random variable, denoted by X, Y,
  etc., is a function or procedure that assigns a
  unique numerical value to each of the outcomes of
  the experiment. The set of possible distinct
  values of the variable is called its Range.

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A Simple Example

• In the coin-tossing experiment described earlier,
  if we let X denote the random variable counting
  the number of H in the outcome, then the values
  of X associated with each of the 8 possible
  outcomes are
• X(HHH) 3, X(HHT) 2, X(HTH) 2, X(THH) 2,
  X(HTT) 1, X(THT) 1, X(TTH) 1, X(TTT) 0
• Therefore, Range of X 0, 1, 2, 3.
• One of the advantages of dealing with random
  variables instead of the elementary events of the
  experiment is that we will be dealing with
  numbers, which we could add, multiply, etc.,
  instead of crude outcomes that we could not do
  arithmetic operations.

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Types of Random Variables

• Random variables can be classified into either
  discrete or continuous variables.
• Discrete variables are those whose range has a
  finite number or at most a countable number of
  values while
• Continuous variables are those whose range
  contains an interval of the real line, hence has
  an uncountable number of values.
• The variable X in the example is clearly a
  discrete random variable.

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Probability Distribution Function

• Definition Given a discrete random variable X
  whose range is R x1, x2, x3, , its
  probability function, denoted by p(x), is a
  table, a graph, or a mathematical formula which
  provides the probabilities for each of the
  possible values in its range. In formal notation,

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Determining the Probability Function

• For a discrete random variable X, the value of
  p(xj) is obtained by summing up the probabilities
  of all the elementary events whose X-value is xj.
  
• A simple illustration makes this immediately
  transparent.
• For the variable X which counts the number of H
  that occur in the toss of three fair coins, we
  obtain the probability function as follows

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Probability Function for X in Example

• The range of X is R 0,1, 2, 3. We have
• p(0) P(X 0) P(TTT) 1/8 .125
• p(1) P(X 1) P(HTT) P(THT) P(TTH) 1/8
  1/8 1/8 3/8 .375
• p(2) P(X 2) P(HHT) P(HTH) P(THH) 1/8
  1/8 1/8 3/8 .375
• p(3) P(X 3) P(HHH) 1/8 .125
• In formula form
• p(x) (3Cx)(1/8), x0,1,2,3.
• In tabular and graphical forms the probability
  function can be presented via

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Tabular/Graphical Presentation of the Probability
Function of X
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Properties and Utilities of Probability Functions

• Note that the sum of the probabilities, which are
  all nonnegative, in a probability function equals
  1. This is so because we take into account all
  the possible outcomes, that is, the sample space,
  and its probability is 1. That is, p(x) satisfies
• p(x) gt 0 for all x, and ?all x p(x) 1.
• The shape of the distribution could be obtained
  from the graph of the probability function.
• For example, the probability distribution for the
  variable X is symmetric with high points at the
  values of 1 and 2.

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Utilities continued

• The probability function of a random variable
  serves as a theoretical model of the population
  of values of the variable, with this population
  being the collection of the outcomes of the
  experiment when it is repeated many, many, many
  times.
• Numerical characteristics of the probability
  function, such as the mean, variance, and
  standard deviation, will be called parameters.
• The probability function can be used to compute
  the probabilities that the variable takes values
  in a certain set of interest.
• For instance, in the example, P(X lt 2) p(0)
  p(1) p(2) 1/8 3/8 3/8 7/8.

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Parameters of a Discrete Probability Distribution

• As mentioned earlier, a probability distribution
  serves as a theoretical model of a population.
• Characteristics of a probability distribution are
  therefore called parameters, and they are usually
  denoted by Greek letters.
• We now discuss four important parameters of
  discrete probability distributions. These are
• Mean (?) and Median (?)
• Variance (?2) and Standard Deviation (?).

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Median of a Discrete Random Variable

• Given a discrete random variable X whose
  probability distribution function is p(x), its
  median, denoted by ?, is any value such that

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A Simple Example

• For the variable X denoting the number of H
  that occur in the toss of three fair coins, we
  therefore have
• Mean ? (0)(1/8)(1)(3/8)(2)(3/8)(3)(1/8)
  12/8 1.5.
• For its median, we could take ? 1.5, since
  notice that P(X lt 1.5) p(0) p(1) 1/8 3/8
  0.5, and also, P(X gt 1.5) p(2) p(3) 1/8
  3/8 0.5.
• However, note that the median is not unique since
  any value of ? between 1 and 2, exclusive, will
  satisfy the definition of being a median.

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Interpretations

• As in the case of the sample statistics we have
  the following interpretations
• The mean, ?, serves as the center of gravity or
  balancing point for the probability
  distribution while
• The median is a value that divides the
  probability distribution into a 5050 split.
• Other properties, like sensitivity of the mean to
  extreme values also holds for these parameters.

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Variance of a Discrete Random Variable

• Given a discrete random variable X taking values
  x1, x2, x3, whose probability distribution is
  p(x), its variance, denoted by ?2, is given by

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Variance continued

• The first formula is called the definitional
  formula, which indicates that the variance is the
  mean of the squared deviations from the mean (?).
• The second formula is called the computational or
  the machine formula, and is easier to implement
  in practice.
• The variance is always nonnegative, and becomes
  zero if and only if the random variable takes
  only one value (we say in this case that it is a
  degenerate variable). The larger the value of the
  variance, the more variability in the
  distribution.
• The variance has squared units of measurements.

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Standard Deviation of a Discrete Random Variable

• Since the variance has squared units of
  measurements, to obtain a measure of variation
  whose units are the same as the variable, we
  define the standard deviation (?) to be the
  positive square root of the variance.
• Formally, it is defined via

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Illustration of Computation of the Variance and
Standard Deviation

• Going back to the variable X which counts the
  number of H in a toss of three fair coins, we
  have, by recalling that ? 1.5 and using the
  definition, that
• ?2 Var(X) (0 - 1.5)2(1/8) (1 - 1.5)2(3/8)
  (2 - 1.5)2(3/8) (3 - 1.5)2(1/8) (2.25)(.125)
  (.25)(.375) (.25)(.375) (2.25)(.125)
  0.75.
• We could also use the computational formula to
  get
• ?2 Var(X) (0)2(.125) (1)2(.375)
  (2)2(.375) (3)2(.125) - (1.5)2 0 .375
  1.5 1.125 - 2.25 3 - 2.25 0.75.
• Therefore, ? StdDev(X) (.75)(1/2) .866.

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Spreadsheet-Type Computation of the Parameters