Lecture 23 continuous random variables

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Lecture 23 continuous random variables Powered By Docstoc
					   Random variables can be classified as either
    discrete or continuous.
   Example:
    ◦ Discrete: mostly counts
    ◦ Continuous: time, distance, etc.
   1. They are used to describe different types
    of quantities.
   2. We use distinct values for discrete random
    variables but continuous real numbers for
    continuous random variables.
   3. Numbers between the values of discrete
    random variable makes no sense, for
    example, P(0)=0.5, P(1)=0.5, then P(1.5) has
    no meaning at all. But that is not true for
    continuous random variables.
   Both discrete and continuous random
    variables have sample space.
   For discrete r.v., there may be finite or
    infinite number of sample points in the
    sample space.
   For continuous r.v., there are always infinitely
    many sample points in the sample space.
   *** For discrete r.v., given the pmf, we can
    find the probability of each sample point in
    the sample space.
   *** But for continuous r.v., we DO NOT
    consider the probability of each sample point
    in the sample space because it is defined to
    be ZERO!
   In another word,
   For discrete random variables, only the value
    listed in the PMF have positive probabilities,
    all other values have probability zero. We can
    find probability for some specific value or an
    interval of values.

   For continuous random variables, the
    probability of every specific value is zero.
    Probability only exists for an interval of
    values for continuous r.v..
   Let X be the number of stops for a citybus
    going from downtown Lafayette to Purdue
    campus. X is a discrete/continuous?
   Let Y be the distance from the train station
    and where a citybus can stop at when it
    comes from downtown Lafayette to Purdue
    campus. Y is a discrete/continuous?
   P(X=3 stops)=?
   P(Y=150 yards)=?
   PDF and CDF.
   PDF is Probability Density Function, it is
    similar to the PMF for discrete random
    variables, but unlike PMF, it does not tell us
    about the probability.
   CDF is Cumulative Distribution Function, it
    has a counterpart for discrete random
    variables, but for continuous random
    variables, it is the only way we can find a
    probability.
   For discrete random variables:
    ◦ PMF: P(X=K)
    ◦ CDF: P(a < X < b) = ∑KP(X=K)
   For continuous random variables:
    ◦ PDF: f(x)
    ◦ CDF: F(x)=P(a < X < b) = ∫ab f(x)dx
   For discrete random variables, both PMF and
    CDF can tell us probabilities.

   For continuous random variables, ONLY CDF
    can tell us probabilities.
   Given X is a continuous random variable with
    sample space Ω and its PDF is f(x), f(x) must
    satisfy the following conditions:
    ◦ 1. 0≤ f(x)
    ◦ 2. ∫Ωf(x) dx= 1
    ◦ The same as the conditions for discrete random
      variables.
   A continuous random variable X has the pdf
    f(x)=c(x-1)(2-x) over the interval [1, 2] and 0
    elsewhere. What value of c makes f(x) a valid
    pdf for X?
   What is P(x>1.5)?
   Think about the citybus example and simplify
    it. Suppose the citybus starts at point A and
    goes toward point B, if this bus can stop at
    will, or stop at each point between A and B
    with equal probability, we let X be the
    distance between where the bus stops and
    point A.
   Then X is a random variable and it is said to
    follow a Uniform distribution.
   We will talk about several continuous
    distributions, we need to know:
    ◦ Their PDF
    ◦ How to calculate probability under those
      distributions.
    ◦ How to find mean and variance for those random
      variables
   For Uniform:
    ◦ PDF:
                       1
                             ,A X  B
             f ( x)   B  A
                      0, elsewhere
                      
   In order to calculate the probability, we need to
    know the distance between A and B.
   In another word, the parameters for a uniform
    distribution are A and B in this case, where A and
    B are defined as the distance mark for the two
    points.
   For example, if B is 2000 yards away from A,
    then B-A=2000.
   And the probability that the bus stops within
    200 yards from A would be


       200                200
                                 1         200
       
       0
             f ( x)dx    
                          0
                                2000
                                     dx 
                                          2000
                                                0.1
   Then what is the probability that the bus
    stops somewhere between 400 yards away
    from A and 600 yards away from A?


     600           600
                        1         200
                    
       f ( x)dx  400 2000 dx  2000  0.1
     400
   What is the probability that the bus stops
    within 200 yards away from point B?


    2000            2000
                           1         200
     
    1800
           f ( x)dx  
                     1800
                          2000
                               dx 
                                    2000
                                          0.1
   What is the probability that the bus stops half
    way between A and B.


     1000                1000
                                 1        1000
      
      0
            f ( x)dx     
                          0
                                2000
                                     dx 
                                          2000
                                                0.5
Given that a continuous r.v. follows a uniform
 distribution with pdf:

               1
                     ,a  X  b
     f ( x)   b  a
              0, elsewhere
              
           ab
  E( X ) 
            2
             (b  a ) 2
 Var ( X ) 
                12
   Let T be the time when a STAT225 student
    turned in his/her exam 1 hour after the exam
    started. Suppose this time is uniformly/evenly
    distributed between 9pm and 9:30pm.
   What is the pdf of T?
   What is the probability that a student turned
    in the exam between 9:10pm and 9:25pm?
   What is the mean and standard deviation of
    T?
   What is the probability that a student turned
    in the exam at 9:30pm?



   What is the probability that a student turned
    in the exam by 9:30pm?

				
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posted:5/22/2013
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