Let X,, X2, .. .be independent Cauchy random variables, each with PDF d f(x)= T(d² +x²)' Show that A, = (X, +X2++X,)/n has the same distribution as X.
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- Suppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?If X1, X2, ... , Xn are independent random variables having identical Bernoulli distributions with the param-eter θ, then X is the proportion of successes in n trials, which we denote by ˆ . Verify that(a) E()ˆ = θ;(b) var()ˆ = θ (1 − θ )n .If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?
- Let X1, X2, ... Xn random variables be independent random variables with a Poisson distribution whose parameters are l1, l2, ... ln, respectively. Which of the following is the moment generating function of the random variable Z defined as (the little image)?(b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If yes, give a proof; if no, give a counterexample.Let X1,...,Xn be iid random variables with expected value 0, variance 1, and covariance Cov [Xi,Xj] = ρ, for i≠j. Use Theorem of linearity of expectation to find the expected value and variance of the sum Y = X1 +...+Xn.
- Let � be a random variabale satisfying �(0,1).Without using the normalcdf(...) function, find �(-3≤�<-2).Consider a random sample X1,...,Xn,... ∼ iid Beta(θ,1) for n > 2. Prove that the MLE and UMVUE are both consistent estimators for θLet X₁,X₂,...,Xₙ denote a random sample from a distribution that is N(0,θ), where the variance θ is an unknown positive number. Show that there exists a uniformly most powerful test of size α for testing the simple hypothesis H₀ : θ = θ', where θ' is a fixed positive number.
- Suppose that Z1, Z2, . . . , Zn are statistically independent random variables. Define Y as the sum of squares of these random variablesLet X1 and X2 be independent chi-squared random variables with r1 and r2 degrees of freedom, respectively. Show that, (a) U = X1/(X1+X2) has a beta distribution with alpha = r1/2 and beta = r2/2. (b) V = X2/(X1+X2) has a beta distribution with alpha = r2/2 and beta = r1/2X is a continuous random variable taking values between 0 and 2. If F(1.5) = 0.70 and P(X<1) = 0.20, what is P (1 < X < 1.5)?