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p73_exercises-3_3

# p73_exercises-3_3 - 3‘3 CUMULATIVE DISTRIBUTION FUNCTIONS...

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Unformatted text preview: 3‘3 CUMULATIVE DISTRIBUTION FUNCTIONS 73 F(x) F(x) 1.0 .—_. 1.000 .__, 0.997 -——o 0.886 :o—o 0.7 0—-—O . 0.2 l i b i 1 ‘ ,5 ‘ —2 O 2 x O 1 2 x Figure 3-3 Cumulative distribution function for Figure 3—4 Cumulative distribution Example 3-7. function for Example 3-8. Figure 3-3 displays a plot of F(x). From the plot, the only points that receive nonzero probability are —2, 0, and 2. The probability mass function at each point is the change in the cumulative distribution function at the point. Therefore, ﬂ—Z) = 0.2 — o = 0.2 f(0) = 0.7 — 0.2 = 0.5 f(2) = 1.0 — 0.7 = 0.3 Suppose that a day’s production of 850 manufactured parts contains 50 parts that do not conform to customer requirements. Two parts are selected at random, without replacement, from the batch. Let the random variable X equal the number of nonconforming parts in the sample. What is the cumulative distribution ﬁmction of X ? I“ The question can be answered by ﬁrst ﬁnding the probability mass ﬁmction of X. 800 799 P(X—0)_ﬁ.849_0'886 800 50 P(X— l)—2'§5—0'%—0.111 50 49 P(X-2)— 850.%—0.003 Therefore, F(O) = P(Xs 0) = 0.886 F(1)= P(Xs l) = 0.886 + 0.111 = 0.997 F(2) = P(Xs 2) = 1 The cumulative distribution ﬁmction for this example is graphed in Fig. 3-4. Note that F(x) is " deﬁned for all x from —<>° < x < 00 and not only for 0, l, and 2. 5 RCISES FOR SECTION 3—3 Determine the cumulative distribution function of the (c) P(-1.1 < X S 1) (d) P(X > 0) um variable in Exercise 3‘14- 3—30. Determine the cumulative distribution function for the '_' . Determine the cumulative distribution function for random variable in Exercise 3-16; also determine the following a . Idom variable in Exercise 3-15; also determine the fol- probabilities: ' g probabilities: (a) P(X < 1.5) (b) P(X s 3) _ P(Xs 1.25) (b) P(Xs 2.2) (c) P(X> 2) . (d) P(l < Xs 2) ...
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