4. Let Xi,X2, , Xn be n i.id. exponential random variables with parameter λ > Let...
Let Xi, X2, X3 be i.id. N(0.1) Suppose Yı = Xi + X2 + X3,Ý, = Xi-X2, у,-X,-X3. Find the joint pdf of Y-(y, Ya, y), using: andom variables. a. The method of variable transformations (Jacobian), b. Multivariate normal distribution properties.
Let Xi,...,Xn be a random sample from a two parameter exponential distribution with pa- rameter θ (λ, μ), (a) Show that the distribution of Ti = log(X(n)-X) +log λ is free of θ. Îs an ancillary statistics (b) show that 72- Xu is ancillary X-X Let Xi,...,Xn be a random sample from a two parameter exponential distribution with pa- rameter θ (λ, μ), (a) Show that the distribution of Ti = log(X(n)-X) +log λ is free of θ. Îs an...
Suppose Xi, X2, . . . , xn are i.id. random variables with Xi ~「α, β). Find the distribution of the sum of the X,'s and the distribution of the average of the X,'s.
Exercise l (Sum of 1.1.d. Exp is Erlang. Let Xi, X2, , Xn ~ Exp(λ) be independent exponential RVs. m Show that fx +x2 (z) = λ2ze-Azi (z 0). (ii) snow that fA+A+x, (z)= 2jaz2e-λΖ1(Z20). (iii) Let Sn-X1 + X2+ + Xn. Use induction to show that Sn ~ Erlang(n, λ), that is, Ctrl+S Ís (z) =-(n-1)!
PROB 4 Let Xi and X2 be independent exponential random variables each having parameter 1 i.e. fx(x) = le-21, x > 0, (i = 1,2). Let Y1 = X1 + X2 and Y2 = ex. Find the joint p.d.f of Yi and Y2.
Let λ >0 and suppose that X1,X2,...,Xn be i.i.d. random variables with Xi∼Exp(λ). Find the PDF of X1+···+Xn. Use convolution formula and prove by induction
suppose Xi, X2, . . . , X, are i.id. random variables with Xi ~ exp(A). Show that Σ-x, ~ T(n, t).
Let X1, X2, ..., Xr be independent exponential random variables with parameter λ. a. Find the moment-generating function of Y = X1 + X2 + ... + Xr. b. What is the distribution of the random variable Y?
4. Let Xi, X2,... be uncorrelated random variables, such that Xn has a uniform distribution over -1/n, 1/n]. Does the sequence converge in probability? 5. Let Xi,X2 be independent random variables, such that P(X) PX--) Does the sequence X1 +X2+...+X satisfy the WLLN? Converge in probability to 0?
Let Xi,X2, , Xn be independent and identically distributed (ii.d.) Exponential(1) random variables. 14] [41 (a) Find the method of moments estimator for X (b) Find the method of moments estimator for (c) Find the bias, variance and MSE (mean square erop) for the essimator in part () Total: [16] Let Xi,X2, , Xn be independent and identically distributed (ii.d.) Exponential(1) random variables. 14] [41 (a) Find the method of moments estimator for X (b) Find the method of moments...