Mathematics for Computer Science

17.3. Distribution Functions 577

3= 36

6= 36

x 2 V

2 3 4 5 6 7 8 9 10 11 12

PDFT.x/

Figure 17.1 The probability density function for the sum of two 6-sided dice.

This is becauseRhas a value for each outcome, so summing the probabilities over
all outcomes is the same as summing over the probabilities of each value in the
range ofR.
As an example, suppose that you roll two unbiased, independent, 6-sided dice.
LetT be the random variable that equals the sum of the two rolls. This random
variable takes on values in the setV D f2;3;:::;12g. A plot of the probability
density function forT is shown in Figure 17.1: The lump in the middle indicates
that sums close to 7 are the most likely. The total area of all the rectangles is 1
since the dice must take on exactly one of the sums inV Df2;3;:::;12g.
Cumulative distribution functions (cdf ’s)are closely-related to pdf’s. The cdf for
a random variableRwhose codomain is a subset of real numbers is the function
CDFRWR!Œ0;1çdefined by:

CDFR.x/WWDPrŒRxç:

As an example, the cumulative distribution function for the random variableT
is shown in Figure 17.2: The height of theith bar in the cumulative distribution
function is equal to thesumof the heights of the leftmostibars in the probability
density function. This follows from the definitions of pdf and cdf:

CDFR.x/DPrŒRxçD

X

yx

PrŒRDyçD

X

yx

PDFR.y/:

In summary, PDFR.x/measures the probability thatR D xand CDFR.x/
measures the probability thatRx. Both PDFRand CDFRcapture the same
information about the random variableR—obviously each one determines the
other —but sometimes one is more convenient. The key point here is that neither
the probability density function nor the cumulative distribution function involves
the sample space of an experiment.