Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
1.6. Discrete Random Variables 45 1.5.9.Consider an urn that contains slips of paper each with one of the num- bers 1, 2 ,...,10 ...
46 Probability and Distributions to obtain the first head. Hence,X(TTHTHHT···) = 3. Clearly, the space ofXis D={ 1 , 2 , 3 , 4 , ...
1.6. Discrete Random Variables 47 correct amperage, the lot is accepted. If, in fact, there are 20 defective fuses in the lot, t ...
48 Probability and Distributions for there are no negative values ofxinDX={x:x=0, 1 , 2 , 3 }. That is, we have the single-value ...
1.7. Continuous Random Variables 49 (b)Show that ∑∞ x=1p(x)=1. (c)DetermineP(X=1, 3 , 5 , 7 ,...). (d)Find the cdfF(x)=P(X≤x). 1 ...
50 Probability and Distributions for some functionfX(t). The functionfX(t) is called aprobability density func- tion(pdf) ofX.If ...
1.7. Continuous Random Variables 51 Taking the derivative ofFX(x), we obtain the pdf ofX: fX(x)= { 2 x 0 ≤x< 1 0elsewhere. (1 ...
52 Probability and Distributions f(x) x 2 4 0.2 0.1 (0, 0) Figure 1.7.1:In Example 1.7.2, the area under the pdf to the right o ...
1.7. Continuous Random Variables 53 which is confirmed immediately by showing thatF′(x)=f(x). For the inverse of the cdf, setu=F ...
54 Probability and Distributions 0.05 0.10 − 8 q 1 q 2 q 3 x f(x) Figure 1.7.2:A graph of the pdf (1.7.9) showing the three quar ...
1.7. Continuous Random Variables 55 a simple formula for the pdf ofYin terms of the pdf ofX, which we record in the next theorem ...
56 Probability and Distributions Example 1.7.6.LetXhave the pdf f(x)= { 4 x^30 <x< 1 0elsewhere. Consider the random varia ...
1.7. Continuous Random Variables 57 F(x) x 1 1 0.5 (0, 0) Figure 1.7.3:Graph of the cdf of Example 1.7.7. Example 1.7.8.Reinsura ...
58 Probability and Distributions 1.7.4.Given ∫ C[1/π(1 +x (^2) )]dx,whereC⊂C={x:−∞<x<∞}. Show that the integral could serv ...
1.7. Continuous Random Variables 59 (b)F(x)=exp{−e−x},−∞<x<∞. (c)F(x)=(1+e−x)−^1 ,−∞<x<∞. (d)F(x)= ∑x j=1 ( 1 2 )j . ...
60 Probability and Distributions 1.7.20.The distribution of the random variableXin Example 1.7.3 is often used to model the log ...
1.8. Expectation of a Random Variable 61 Definition 1.8.1(Expectation).LetXbe a random variable. IfXis a continuous random varia ...
62 Probability and Distributions Remark 1.8.1.The terminology of expectation or expected value has its origin in games of chance ...
1.8. Expectation of a Random Variable 63 Because ∑ x∈SX|g(x)|pX(x) converges, it follows by a theorem in calculus (^6) that any ...
64 Probability and Distributions Proof:For the continuous case, existence follows from the hypothesis, the triangle inequality, ...
«
1
2
3
4
5
6
7
8
9
10
»
Free download pdf