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  1. H

    calculating $E(Y)$

    suppose X is random variable non negative with Distribution function F_{X}(x) and Y is random variable with Distribution function G_{Y}(t)=1-E(e^{-tX}) ,0\leq t . how can i calculate E(Y)
  2. H

    a question in uniform distribution

    suppose X1,…,Xn be random sample of U(0,θ) distribution. if X(n) be largest order statistics of sample, and V=nX¯/X(n), how can I calculate Var(3V−X(n)) X¯ is mean of sample.
  3. H

    A question in order statistics of continuous type distribution

    Let X1,X2,… be a sequence of random variables from a continuous type distribution and m and n be two integers such that m<n, and 2≤n−m. How can I show the probability that the third-order statistic of X1,…,Xm is equal to the fifth-order statistic of X1,…,Xn is 6⋅(n−5 m−3)(n m)?
  4. H

    order statistics of random sample

    suppose X1,X2,…,Xn is a random sample of distribution with positive values where E(X)=Var(X)=1. We show order statistics of this random sample with Y1,Y2,…,Yn. How can show 1. E(∑(Yi/Xi))≥n 2. E(∑YiXi)≤n+n2
  5. H

    distribution total number of car accidents in a lifetime of the car

    the number of car accidents every year has Poisson distribution with mean λ and lifetime of the car has exponential distribution with mean 1/μ. how can find distribution total number of car accidents in a lifetime of the car?
  6. H

    finding E(N) in this question

    suppose X1,X2,… is sequence of independent random variables of U(0,1) if N=min{n>0:X_(n:n)−X_(1:n)>α,0<α<1} that X_(1:n) is smallest order statistic and X_(n:n) is largest order statistic. how can find E(N)
  7. H

    finding UMVUE of parameter 1/(1+λ)

    suppose X1,X2,X3 are a random sample of exponential distribution with parameter λ. how can i find UMVUE parameter 1/(1+λ). note: (T=∑Xi,(i=1,2,3); f(x)=λexp(−λx)). I know that T= X1+X2+X3 is the minimal sufficient statistic and is complete and that i want g(t), where ∫_0^∞(g(t))((λ^3)/2)...
  8. H

    finding UMVU estimator of θ^2

    suppose X_1,…,X_n be a random sample of distribution with probability density function f(x)=2/θ *x* exp(−(x^2)/θ),x>0. how can i find UMVU estimator of θ^2
  9. H

    finding UMVU estimator of parameter γ(θ)

    suppose X_1,X_2,…,X_n,X_{n+1} be a random sample of distribution N(θ,1). if γ(θ)=P(∑Xi>X_{n+1}) (i=1,2,...,n) how can i find UMVU estimator of parameter γ(θ)
  10. H

    bayes estimator

    Let x1,x2,…,xn be a random sample from the normal distribution N(θ,σ^2), where θ has distribution N(μ,τ^2) and where σ^2, μ and τ^2 are known. Find a Bayes estimator δ of θ under the loss function L(θ,δ)=1−exp(−(δ−θ)^2/2γ^2) where γ is known
  11. H

    unbiased confidence interval

    Hi let x1,x2,...,xn be a random sample from distribution in under function distribution . F(x)=[(x/θ)]^β , 0≤x<θ where β is unknown and θ is known find a unbiased confidence interval in level (1-α) for β
  12. H

    shortest confidence interval

    Hi let x1,x2,...,xn be a random sample from distribution in under function distribution . F(x)=[(x/θ)]^β , 0≤x<θ if β is unknown and θ is known find a shortest confidence interval in level (1-α) for β^2
  13. H

    π-system and monotone class

    Assume P is a π-system (that is, P is closed under finite intersections) and M is monotone class (that is M is a non-empty collection of subsets of Ω closed under monotone limits). Show that P⊂M does not imply σ(P)⊂M.
  14. H

    sigma -fieled

    Let X : \Omega \rightarrow R and F be a class of subsets of R. show \sigma(X^{-1} (F))=X^{-1}(\sigma(F)) where X^{-1} (F)={X^{-1} (A):A\in F}