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- Suppose Y1,Y2,··· ,Yn are i.i.d. continuous uniform(0,1) distributed.(a) Prove that the kth-order statistic, Y(k), has Beta distribution with α = k and β = n−k+ 1.(b) What is the distribution of the median of Y1,Y2,··· ,Yn when n is an odd integer?(c) Find the joint density of the middle two of Y(1),Y(2),··· ,Y(n) when n is an even integer2. Let X~ f(x; 0) for 0 € © and suppose X = {x : f(x; 0) > 0} does not depend on 0. If 2² дөдх f(x; 0). :ƒ(x; 0) > ə 20 ə f(x; 0)ƒ(x;0) for almost every (x,0) = X × 0, then ƒ(x; 0) has monotone likelihood ratio in x.b) Suppose that X is a random variable with the probability density function given by f(x) = {2(1- (2(1-x), 0≤x≤1 otherwise Find the density function of W = 2X - 1 using method of cumulative distribution function.
- 34. Consider the continuous random variable X whose pdf is given by f(r) = (a-r³)Io<<1(1). (a) Find the value of a that makes f(z) a pdf. (b) Find E(X). (c) Find V(X). (d) Find P(X ≤).1. X is a normally distributed variable with mean μ = 30 and standard deviation o = 4. Find a) P(x 21) c) P(30Consider the Weibull distribution with pdf fx (x) = 3x²e¬**, x > 0. Derive the Hazard rate function (failure rate) fx (t) R (t) : %3D 1- Fx (t)3. A random variable X has a cumulative distribution function if 0 0.5). (e) Find P(X < 1.25). (f) Find P(X = 1.25).b) Let X₁, X2, X3,...,Xn be a random sample of n from population X distributed with the following probability density function: f(x;0)=√√2n0 0, -20₁ if -∞0 < x <∞0 otherwise (i) Find the parameter space of 0. (ii) Find the maximum likelihood estimator of 0. (iii) Check whether or not the estimator obtained in (ii) is unbiased. (iv) Find the Fisher information in this sample of size n about the parameter 0.If Y = InX has a N (0, 1) distribution then E (X²) is equal to