DISCRETE PROBABILITY DISTRIBUTION


Discrete probability density function: probability assigned to AREA of discrete random variable cells .

Probability ( Cell area) = Height (X k ) · Width (W c )

Discrete probability is the frequency of the grouped values x = X k for the data corresponding to the individual event S i in sample space. This may be shown in a pictorial form in Figure 16.1.


Figure 16.1: Discrete probability density function.

However, when data are grouped as consecutive integer values the cell width is unity; W c = 1.

  1. Probability of group data with cell width W c :

    P(x = X k ) = f (X k ) · W c ; k = 1, 2, 3, ..., K

    where f(X k ) is the probability density for the RV cell X k , which has a sample frequency f k for a total sample size of n:

    f(X k ) = f k /n

  2. Discrete probability density function properties:

    1. Positive 0 < P (X k ) = f(X k ) · W c

    2. Unit area (area is sum under curve unity)

RANDOM EXPERIMENT

Two tosses of a coin.

Random variable of sample event X i is defined as the number of heads to appear in two tosses.

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

  • SAMPLE SPACE: TT TH HT HH

  • Probability (sample space): 1/4 1/4 1/4 1/4

  • Random Variable X i : 0 1 1 2 ascending (Number of Heads)

  • Grouping into four cells each of width, W k = 1;

  • Discrete probability density function f(X k ):

    f(0) = 1/4; f(1) = 1/4 + 1/4 = 1/2; f(2) = 1/4

The above may be represented in graphic displays as in Figure 16.2.

click to expand
Figure 16.2: A bar chart and a histogram of two tosses of a coin.



Six Sigma and Beyond. Statistics and Probability
Six Sigma and Beyond: Statistics and Probability, Volume III
ISBN: 1574443127
EAN: 2147483647
Year: 2003
Pages: 252

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