Z1,Z2, ... are iid uniformly distributed on [-1,1]. Prove that nSn N(0,1) on Lyapunov condition.
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- Suppose that we observe that X1, X2, . . . , Xn are iid∼ U(0, 1). Show that X(1)converges in probability to zero.Suppose that X is uniformly distributed on [0, 4]. Define Y = 2X + 5. Compute the pdf and cdf of Y.Suppose X has uniform distribution on (−1, 1) and, given X = x, Y is uniformly distributed on (−(1-x^2)^1/2, (1+x^2)^1/2). Find the join denisty of X and Y.
- If X, Y are normally distributed independent random variables, then show that W = 2X - Y is normally distributed. Do not use moment generating function, only use convolution formula.The 4-dimensional random vector X has PDF fX(x)={1 when 0≤xi≤1, i=1,2,3,4 and 0 otherwise}. Are the for components of X independent random variables?Let XXX and YYY be uniformly distributed on [0,1][0,1][0,1] and independent. What is the expectation of the product XYXYXY, i.e., E[XY]E[XY]E[XY]? Provide at least two decimal places.
- Let {Xn} and {Yn} be sequences of random variables such that Xn diverges to ∞ in probability and Yn is bounded in probability. Show that Xn +Yn diverges to ∞ in probability.Find the density of Z = X-Y for independent Exp(λ) random varibles X and Y. Show all Work!!Prove that cov(X, Y) = cov(Y, X) for both discreteand continuous random variables X and Y.
- X and Y are continuous random variables with pdf f(x,y) = 2x for0 ≤x ≤y ≤1, and f(x,y) = 0 otherwise. Find the conditional expectation ofY given X = x.Let X bea randomvariable having the uniformdistribution onthe interval(θ,θ+1),θ∈R.Show thatX isnot complete.Prove that the rv Y=(tan^-1(X1/X2)) and Z=(X1^2+X2^2)^0.5 are independent. No distribution is given.