082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

29.3 Probability distributions -- discrete variable 935
29.3 PROBABILITYDISTRIBUTIONS--DISCRETEVARIABLE
The range of values that a variable can take does not give sufficient information about
the variable. We need to know which values are likely to occur often and which values
willoccuronlyinfrequently.Forexample,supposexisadiscreterandomvariablewhich
cantakevalues0,1,2,3,4,5and6.Wemayaskquestionssuchas‘Whichvalueismost
likelytooccur?’,‘Isa6morelikelytooccurthana5?’,andsoon.Weneedinformation
on the probability of each value occurring. Suppose that information is provided and is
given in Table 29.1. Ifxis sampled 100 times then on average 0 will occur 10 times, 1
will occur 10 times, 2 will occur 15 times and so on. Table 29.1 is called aprobability
distribution for the random variable x. Note that the probabilities sum to 1; the table
tells us how the total probability isdistributed among the various possible values of the
random variable. Table 29.1 may be represented ingraphical form (see Figure 29.1).
P(x)
0.3
0.2
Table29.1
The probability of a discrete
value occurring.
x 0 1 2 3 4 5 6
P(x) 0.1 0.1 0.15 0.3 0.2 0.1 0.05
0.1
0 1 2 3 4 5 6
Figure29.1
Plotted data ofTable 29.1.
x
EXERCISES29.3
1 Theprobabilitydistribution forthe randomvariable,
x,is
x 2 2.5 3.0 3.5 4.0 4.5
P(x) 0.07 0.36 0.21 0.19 0.10 0.07
(a) StateP(x = 3.5)
(b) CalculateP(x 3.0)
(c) CalculateP(x < 4.0)
(d) CalculateP(x > 3.5)
(e) CalculateP(x 3.9)
(f) The variable,x,is sampled50000 times. How
many timeswould you expectxto have a value
of2.5?
2 The probabilitydistribution ofthe randomvariable,y,
is given as
y −3 −2 −1 0 1 2 3
P(y) 0.63 0.20 0.09 0.04 0.02 0.01 0.01
Calculate
(a) P(y 0) (b) P(y 1)
(c) P(|y| 1) (d) P(y 2 > 3)
(e) P(y 2 < 6)
Solutions
1 (a) 0.19 (b) 0.57 (c) 0.83 (d) 0.17
(e) 0.83 (f) 18000
2 (a) 0.08 (b) 0.98 (c) 0.15 (d) 0.85
(e) 0.36