Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
1.4. Conditional Probability and Independence 25 hand (A), is, sinceA∩B=B, P(B|A)= P(B) P(A) = ( 13 5 ) / ( 52 5 ) [( 13 4 )( 39 ...
26 Probability and Distributions ButP(A∩B)=P(A)P(B|A). Hence, providedP(A∩B)>0, P(A∩B∩C)=P(A)P(B|A)P(C|A∩B). This procedure c ...
1.4. Conditional Probability and Independence 27 Example 1.4.5.Say it is known that bowlA 1 contains three red and seven blue ch ...
28 Probability and Distributions Example 1.4.7.Suppose we want to investigate the percentage of abused children in a certain pop ...
1.4. Conditional Probability and Independence 29 Definition 1.4.2.LetAandBbe two events. We say thatAandBareinde- pendentifP(A∩B ...
30 Probability and Distributions In particular, ifA 1 ,A 2 ,...,Anare mutually independent, then P(A 1 ∩A 2 ∩···∩An)=P(A 1 )P(A ...
1.4. Conditional Probability and Independence 31 Example 1.4.11.A computer system is built so that if componentK 1 fails, it is ...
32 Probability and Distributions at 0. The commandsample(rngA,1,pr=pA)draws a sample of size 1 fromrngA with pmfpA. Each executi ...
1.4. Conditional Probability and Independence 33 1.4.3.Suppose we are playing draw poker. We are dealt (from a well-shuffled dec ...
34 Probability and Distributions 1.4.11.SupposeAandBare independent events. In expression (1.4.6) we showed thatAcandBare indepe ...
1.4. Conditional Probability and Independence 35 (b)Letpdenote the probability of a 6. Show that the game favors Bob, for allp, ...
36 Probability and Distributions 1.4.27.The following game is played. The player randomly draws from the set of integers{ 1 , 2 ...
1.5. Random Variables 37 light smokers were five and three times that of the nonsmokers, respectively. A ran- domly selected par ...
38 Probability and Distributions As Exercise 1.5.11 shows,PX(D) is a probability onD. An example is helpful here. Example 1.5.1 ...
1.5. Random Variables 39 there are some intuitive probabilities. For instance, because the number is chosen at random, it is rea ...
40 Probability and Distributions F(x) x 1 2 3 4 5 6 1.0 0.5 (0, 0) Figure 1.5.1:Distribution function for Example 1.5.3. The fol ...
1.5. Random Variables 41 F(x) x 1 1 (0, 0) Figure 1.5.2:Distribution function for Example 1.5.4. The cdfs displayed in Figures 1 ...
42 Probability and Distributions Proof:Note that {−∞<X≤b}={−∞<X≤a}∪{a<X≤b}. The proof of the result follows immediately ...
1.5. Random Variables 43 which is the desired result. Example 1.5.6.LetXhave the discontinuous cdf FX(x)= ⎧ ⎨ ⎩ 0 x< 0 x/ 20 ...
44 Probability and Distributions EXERCISES 1.5.1.Let a card be selected from an ordinary deck of playing cards. The outcome cis ...
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