Variance and Standard 
      Deviation
          Example:   Variance and Standard Deviation of the Number of 
                  Radios Sold in a Week
          x, Radios    p(x), Probability     (x -  )2  p(x)
                                                  X
           0           p(0) = 0.03      (0 – 2.1)2 (0.03) = 0.1323
           1           p(1) = 0.20      (1 – 2.1)2 (0.20) = 0.2420
           2           p(2) = 0.50      (2 – 2.1)2 (0.50) = 0.0050
           3           p(3) = 0.20      (3 – 2.1)2 (0.20) = 0.1620
           4           p(4) = 0.05      (4 – 2.1)2 (0.05) = 0.1805
           5           p(5) = 0.02      (5 – 2.1)2 (0.02) = 0.1682
                                  1.00                                0.8900
    µx =                                             Standard deviation
        2.10 Variance
                                                     X  0.89 0.9434
                2 0.89
                   X
   Expected Value and Variance 
   (Summary)
      The expected value, or mean, of a random variable
      The expected value, or mean, of a random variable
      is a measure of its central location.
      is a measure of its central location.
      The variance summarizes the variability in the
      The variance summarizes the variability in the
      values of a random variable.
      values of a random variable.
      The standard deviation, , is defined as the positive
      The standard deviation, , is defined as the positive
      square root of the variance.
      square root of the variance.
    Expected Value and Variance 
    (Summary)
    The expected value, or mean, of a random 
       variable is a measure of its central 
       location.        E(x) =  = xf(x)
                        E(x) =  = xf(x)
    The variance summarizes the variability in 
       the values of a random variable.
    The standard deviation, is defined as the 
       positive  square root of the variance.
                                 2           2
                                 2           2
                     Var(x) =    = (x - ) f(x)
                     Var(x) =    = (x - ) f(x)
                  Discrete Probability 
                  Discrete Probability 
                  Distribution Models
                  Distribution Models
                                                   Discrete
                                                Probability
                                               Distribution
          Binomial                    Hyper-                   Negative                 Poisson
                                   Geometric                  Binomial
    Binomial Distribution
     Four Properties of a Binomial Experiment
         1.  The experiment consists of a sequence of n
          1.  The experiment consists of a sequence of n
              identical trials.
               identical trials.
         2.  Two outcomes, success and failure, are possible
          2.  Two outcomes, success and failure, are possible
              on each trial.
               on each trial.
         3.  The probability of a success, denoted by p, does
          3.  The probability of a success, denoted by p, does
              not change from trial to trial.
               not change from trial to trial.
                                              stationarit
         4.  The trials are independent.           y
          4.  The trials are independent.     assumptio
                                                   n