let X=pareto(α,γ) find the distribution and density function of Y=logX
Let X=pareto(α,γ) find the distribution and density function of Y=logX
b. Suppose ~ Γ(α, β), with α > 0, β > 0 and let Y-eu. Find the probability density function of Y Find EY and var(Y)
Q1. Assume that X is Pareto random variables with the density -α-1 , r21, where α > 0 (a) Calculate EX]. What do you need to assume about a for E[X to be finite? (b) Find the density of X + b for b 〉 0. (c) Find the cumulative distribution function of Y log X.
If X follows a two-parameter Pareto distribution with a = 3 and θ = 100, find the density function of Y, where Y- 1.5X
The random variable X is distributed as a Pareto distribution with parameters α = 3, θ. E[X] = 1. The random variable Y = 2X. Calculate V ar(Y )
Let 0 < γ < α . Then a 100(1 − α )% CI for μ when n is large is Xbar+/-zγ*(s/sqrt(n))The choice γ = α /2 yields the usual interval derived in Section 8.2; if γ ≠ α /2, this confidence interval is not symmetric about . The width of the interval is W=s(zγ+ zα-γ)/sqrt(n). Show that w is minimized for the choice γ = α /2, so that the symmetric interval is the shortest. [ Hints : (a)...
Question 8. Let X be the Exponential distribution with parameter 2. Let Y=A7. a) Find the distribution function of Y. b) Find the density function of Y. c) Find the distribution of Y.
Verify that the probability density function of Γ(α,λ) inte- grates to 1.
Let X1, . . . , Xn be a sample taken from the Gamma distribution Γ(2, θ−1) with pdf f(x,θ)= θ^2xexp(−θx) if x ≥ 0, θ ∈ (0,∞), and 0 otherwise, (A) Show that Y = ∑ni=1 Xi is a complete and sufficient statistic. (B) Find E(1/Y) . Hint: If W ∼ χ2(k) then E(W^m) = 2mΓ(k/2+m) for m > −k/2. Note also that Y Γ(k/2) Γ(n) = (n − 1)!, n ∈ N∗ . Facts from 1(C) are useful:...
The Pareto distribution has been used in economics as a model for a density function with a slowly decaying tail: f(x|x0, θ) = θx0θ x−θ−1, x ≥ x0,θ > 1 Assume that x0 > 0 is given and that X1,...,Xn is an i.i.d. sample. find a sufficient statistic for θ
7.4 Let X ~ U(-1,1) and Y = x2. a. What are the density, the distribution function, the mean, and the variance of Y: b. What is Pr[Y < 0.5]? 7.5 Let X – U(0,1), and let Y = eax for some a > 0. What are the density, the distribution function, the mean, and the variance of Y?