Six Sigma and Beyond: Statistics and Probability, Volume III

RANDOM VARIABLES GROUPED INTO CELLS

  1. Group random variables into cells (or " bins ") if the number of random variables is large, say M > 30.

  2. Groups or cells are formed in the numerical RV domain by dividing the range into say 10 to 20 equal cells.

  3. While the total number of RVs is M, the number of cells is K.

  4. All the individual RVs in a cell assume the value of the RV in, say the center of the cell interval and are denoted X k .

  5. Each cell or grouping is designated X k and represents a finite interval of contiguous RVs.

    Individual:

    X i = [(X 1; X 2 , ...), ..., [X i-1 , ..., (X M-1 , X M )]

    Cells:

    X k = [(X 1 ), ..., (X k ), ..., [X K )]

    where X k = X i-1 ‰ X i < X ik

    This may be shown as:

RANDOM EXPERIMENT

Two tosses of a coin.

Sample Space: S 1 = TT, S 2 = TH, S 3 = HT, S 4 = HH

Function Process:

  1. Assign real-value number to occurrence: T = 0 and H = 1.

  2. Sum of real-values for individual samples S i is RV X i .

    S 1 = TT ’ X 1 = 0 + 0 = 0

    S 2 = TH ’ X 2 = 0 + 1 = 1

    S 3 = HT ’ X 3 = 1 + 0 = 1

    S 4 = HH ’ X 4 = 1 + 1 = 2

    Random Variables: arranged in ascending order by magnitude

    {S i } = {TT, TH, HT, HH} ’ {X i } = {0, 1, 1, 2}

    Note  

    cell width W c = 1.

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