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An Introduction to Mathematical Statistics and Its Applications (6th Edition)
- Given the probability mass function of a random variable X p(x) = c/(x+1) if x = 0,1 and p(x) = 0 in other mass points. What is the value of c ?arrow_forwardLet X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0arrow_forwardIf we let RX(t) = ln MX(t), show that R X(0) = μ and RX(0) = σ2. Also, use these results to find the mean and the variance of a random variable X having the moment-generating function MX(t) = e4(et−1)arrow_forward
- Suppose we have the quadratic function f(x)=A(x^2)+C where the random variables A and C have densities fA(x)=(x/2) for 0≤x≤2, and fC(x)=3(x^2) for 0≤x≤1. Assume A and C are independent. Find the probability that f(x) has real roots.arrow_forwardLet X1, . . . , Xn be iid with pdf f(x) = 1 x √ 2πθ2 e − (log(x)−θ1) 2 2θ2 , −∞ < x < ∞, and unknown parameters θ1 and θ2. Find the maximum likelihood estimators for θ1 and θ2, respectivelyarrow_forwardf X1,X2,...,Xn constitute a random sample of size n from a geometric population, show that Y = X1 + X2 + ···+ Xn is a sufficient estimator of the parameter θ.arrow_forward
- Find the maximum likelihood estimator for θ in the pdf f(y; θ) = 2y/(1 − θ^2), θ ≤ y ≤ 1.arrow_forward1)Let x be a uniform random variable over the interval (0, 1). Knowing that y = x2 , calculate:a)Determine Fy(Y) = P(y<=Y),Y real and determine the pdf of y.b)Calculate E[x2] , using the pdf of x.c)Calculate E[y], using the pdf of y and compare with part (b).arrow_forwardSuppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise c=1/2 P(X < 1), Determine whether X and Y are independentarrow_forward
- Sketch the ensemble, that is, realizations of the random process X(t) = A cos(2πft), where f is a uniform random variable U(200, 300). That is, f is uniformly distributed in the range [200, 300] Hz and A = 7 is a constant.arrow_forwardIf the probability density of X is given by f(x) =kx3(1 + 2x)6 for x > 00 elsewhere where k is an appropriate constant, find the probabilitydensity of the random variable Y = 2X 1 + 2X . Identify thedistribution of Y, and thus determine the value of k.arrow_forwardSuppose the joint probability density of X and Y is fX,Y (x, y) = 3y 2 with 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 and zero everywhere else. 1. Compute E[X|Y = y]. 2. Compute E[X3 + X|X < .5]arrow_forward
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