Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
1.8. Expectation of a Random Variable 65 1.8.1 R Computation for an Estimation of the Expected Gain In the following example, we ...
66 Probability and Distributions } } return(gain) } The following R script obtains the average gain for a sample of 10,000 games ...
1.8. Expectation of a Random Variable 67 1.8.8.A bowl contains 10 chips, of which 8 are marked $2 each and 2 are marked $5 each. ...
68 Probability and Distributions 1.9 SomeSpecialExpectations Certain expectations, if they exist, have special names and symbols ...
1.9. Some Special Expectations 69 It is customary to callσ(the positive square root of the variance) thestandard deviationofX (o ...
70 Probability and Distributions Example 1.9.3.IfXhas the pdf f(x)= { 1 x^21 <x<∞ 0elsewhere, then the mean value ofXdoes ...
1.9. Some Special Expectations 71 Theorem 1.9.2.LetXandY be random variables with moment generating func- tionsMXandMY, respecti ...
72 Probability and Distributions us to interchange the order of differentiation and integration (or summation in the discrete ca ...
1.9. Some Special Expectations 73 Example 1.9.5.It is known that the series 1 12 + 1 22 + 1 32 +··· converges toπ^2 /6. Then p(x ...
74 Probability and Distributions consider this alternative method. The functionM(t) is represented by the following Maclaurin’s ...
1.9. Some Special Expectations 75 Every distribution has a unique characteristic function; and to each charac- teristic function ...
76 Probability and Distributions 1.9.7.Show that the moment generating function of the random variableXhaving the pdff(x)=^13 ,− ...
1.9. Some Special Expectations 77 (a)f(x)=^12 ,− 1 <x<1, zero elsewhere. (b)f(x)=3(1−x^2 )/ 4 ,− 1 <x<1, zero elsewh ...
78 Probability and Distributions 1.9.23.LetXhave the pmfp(x)=1/k, x=1, 2 , 3 ,...,k, zero elsewhere. Show that the mgf is M(t)= ...
1.10. Important Inequalities 79 which is the the desired result. Theorem 1.10.2(Markov’s Inequality).Letu(X)be a nonnegative fun ...
80 Probability and Distributions Proof.In Theorem 1.10.2 takeu(X)=(X−μ)^2 andc=k^2 σ^2 .Thenwehave P[(X−μ)^2 ≥k^2 σ^2 ]≤ E[(X−μ) ...
1.10. Important Inequalities 81 Definition 1.10.1.Afunctionφdefined on an interval(a, b),−∞ ≤a<b≤∞, is said to be aconvexfunc ...
82 Probability and Distributions Example 1.10.3.LetXbe a nondegenerate random variable with meanμand a finite second moment. The ...
1.10. Important Inequalities 83 1.10.2.LetXbe a random variable such thatP(X≤0) = 0 and letμ=E(X) exist. Show thatP(X≥ 2 μ)≤^12. ...
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