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.

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