Fundamentals of Probability and Statistics for Engineers

Consider now the conditional mass function pX Y (xy). With Y y having
happened, the situation is again similar to that for determining pY (y) except
that the number of cars available for taking possible eastward turns is now
n y; also, here, the probabilities p and r need to be re normalized so that they
sum to 1. Hence, pX Y (xy) takes the form

Finally, we have pX Y (x,y) as the product of the two expressions given by
Equations (3.51) and (3.52). The ranges of values for x and y are x 0,1,...,
n y,and y 0,1,...,n.
Note that pX Y (x,y) has a rather complicated expression that could not have
been derived easily in a direct way. This also points out the need to exercise care
in de te rmining the limits of validity for x and y.

Ex ample 3. 10. Problem: resistors are designed to have a resistance R of
50 2. Owing to imprecision in the manufacturing process, the actual density
function of R has the form shown by the so lid curve in Figure 3.18. Determine
the density function of R after screening – that is, after all the resistors having
resistances beyond the 48–52 range are rejected.
Answer: we are interested in the conditional density function, fR(rA), where
A is the event. This is not the usual co nditional den sity function
but it can be found from the basic definition of conditional probability.
We start by considering

48 50 52

fR

fR(r\A)

fR(r)

r

Figure 3. 18 The actual, fR(r), and conditional, fR(rA), for Example 3.10

Random Variables and Probability D istributions 65

j ˆ



j

pXY…xjy†ˆ
ny
x

p

r‡p

x

1 

p

r‡p

nyx

; xˆ 0 ; 1 ;...;ny;yˆ 0 ; 1 ;...;n:

… 3 : 52 †

ˆ

ˆ



j

f 48 R 52 g

FR…rjA†ˆP…Rrj 48 R 52 †ˆ

P…Rr\ 48 R 52 †

P… 48 R 52 †

:

j

(Ω)