Random Variable

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Chapter 2

• Random Variable their Distribution

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Illustration
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Definition

• R.V say X is a function defined over a sample
  space S, that associates a real number, X(e)x,
  whith each possible outcome e in S

Look at example above!
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Other Example

• An experiments involving a sequence of 5 tosses
  of a coin, the number of Heads in the sequence is
  a random variable
• Two rolls of a die, v.r
• ?The sum of the two rolls
• ?The number of sixes in the two rolls
• ?The second roll raised to fifth power

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Main Concepts Related to RV

• A RV is a real valued function of the outcome of
  the experiment
• A function of R.V defines another R.V
• A R.V can be conditioned on an event or on
  another R.V
• There is a notion of independence of a R.V from
  an event or from another R.V

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Definition

• let us consider functions which take values in
  the real numbers.
• In the coin tossing example, our function might
  count the number of heads. Call this function R.
  We can look at the set
• If we have chosen the set of events to contain
  all subsets of ?, then this set is an event, and
  we can ask for the probability of R6
• The precise relation is that if the model is
  (?,F,Pr) and R??(-?,?) then for every interval
  I,
• R?Iw??R(w)?I?F 

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• Definition
• Function which satisfied
• Are called (real valued) R.V

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• Example 1

R is a R. V since for every interval I the set
R?I is a subset of ?, and all subsets of ? are
in F
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Example 2
R is not a R.V since R22 is not in F
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Discrete R.V

• R.V si discrete if its range is finite or at most
  countably infinite
• Definition
• If the set of all possible values of a R.V X
  is a countable set,
• then X called a discrete R.V
• ?f(x)PXx,
• called the discrete probability density function
  (discrete pdf)

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Definition
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Example 1
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Example 2

• The experiment consist of two independent tosses
  of a fair coin, let X be the number of heads
  obtained, then the pdf of X is

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Example 3

• If
• Then find c!

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EXERCISE
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Cummulative Density Function
Definition
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Theorem

• A function F(x) is a CDF for some R.V X if and
  only if it satisfies the following properties

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Example 1
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Example 2

• Suppose that a days production of 850
  manufactured parts contains 50 parts that dont
  conform to customer requirements. Two parts are
  selected at random, without replacement, from the
  batch. Let the random variable X equal the number
  of nonconforming parts in the sample. What the
  cdf of X?

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Exercise