By using a change of variables, verify that the univariate Gaussian distribution given by 1 1 N(x|μ, σ²) = (200²)1/2 exp{- 202 (x − μ)²} 20² satisfy ∞ E[x] = N(x|μ₁0²)xdx = μ -∞

If X has a standard normal distribution (X ∼ N(0, 1)), then the characteristic function of X is ϕX(t) = e −0.5(t^2) . Using this fact, show that X has skewness 0 and kurtosis 3

Let X1, , Xµ be iid with population density (1 0) I>0, Sx(x) = %3D otherwise. Here 0 is an unkown population parameter. 0 has an Exponential(1) distribution. Find the method of moment estimator for 0. Let's call this 6. Is ô unbiased for 0 ? Explain with precise computation. Show that X Find the maximum likelihood estimator for 0. Let's call this 62. Is ô2 unbiascd for 0 ? Explain with precise computation.

Suppose Xn is the sample mean of a random sample of size n from a distribution that has a Gamma(2,1/2) (where a = 2, B = }) pdf f(x) = 4xe-2a, 0 < x < ∞, zero elsewhere. Use the Central Limit Theorem to deduce that the random variables Vn(Xn – 1) converges in distribution to N(0, 1/2).