Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
3.1. The Binomial and Related Distributions 165 3.1.18.One way of estimating the number of fish in a lake is the followingcaptur ...
166 Some Special Distributions 3.1.21.LetX 1 andX 2 have a trinomial distribution. Differentiate the moment- generating function ...
3.2. The Poisson Distribution 167 3.1.29.Let the independent random variablesX 1 andX 2 have binomial distri- butions with param ...
168 Some Special Distributions that is,p(x) satisfies the conditions of being a pmf of a discrete type of random variable. A ran ...
3.2. The Poisson Distribution 169 Integrating both side with respect tot,wehaveforsomeconstantcthat logg(0,t)=−λt+c or g(0,t)=e− ...
170 Some Special Distributions for all real values oft.Since M′(t)=eλ(e t−1) (λet) and M′′(t)=eλ(e t−1) λetλet+eλ(e t−1) λet the ...
3.2. The Poisson Distribution 171 The Poisson distribution satisfies the following important additive property. Theorem 3.2.1.Su ...
172 Some Special Distributions 3.2.5.LetXhave a Poisson distribution withμ= 100. Use Chebyshev’s inequality to determine a lower ...
3.3. TheΓ,χ^2 ,andβDistributions 173 3.2.14.LetXhave a Poisson distribution. IfP(X=1)=P(X= 3), find the mode of the distribution ...
174 Some Special Distributions Ifα>1, an integration by parts shows that Γ(α)=(α−1) ∫∞ 0 yα−^2 e−ydy=(α−1)Γ(α−1). (3.3.1) Acc ...
3.3. TheΓ,χ^2 ,andβDistributions 175 and M′′(t)=(−α)(−α−1)(1−βt)−α−^2 (−β)^2. Hence, for a gamma distribution, we have μ=M′(0) = ...
176 Some Special Distributions 0 5 10 15 0.00 0.15 0.30 x f(x ) α = .25 α = .5 α = 1 β = 4 0 5 10 15 20 25 30 35 0.00 0.06 0.12 ...
3.3. TheΓ,χ^2 ,andβDistributions 177 One of the most important properties of the gamma distribution is its additive property. Th ...
178 Some Special Distributions That is, the waiting time until thekthevent,Wk, has the gamma distribution with α=kandβ=1/λ.LetT ...
3.3. TheΓ,χ^2 ,andβDistributions 179 command plots the density function over the interval (0,24): x=seq(0,24,.1);plot(dchisq(x,1 ...
180 Some Special Distributions Because theχ^2 -distributions are a subfamily of the Γ-distributions, the additiv- ity property f ...
3.3. TheΓ,χ^2 ,andβDistributions 181 In accordance with Theorem 2.4.1 the random variables are independent. The marginal pdf ofY ...
182 Some Special Distributions Let Yi= Xi X 1 +X 2 +···+Xk+1 ,i=1, 2 ,...,k, andYk+1=X 1 +X 2 +···+Xk+1denotek+1 new random vari ...
3.3. TheΓ,χ^2 ,andβDistributions 183 3.3.3.Suppose the lifetime in months of an engine, working under hazardous con- ditions, ha ...
184 Some Special Distributions 3.3.10.Give a reasonable definition of a chi-square distribution with zero degrees of freedom. Hi ...
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