2. Let YBinomial (n, p) where p docs not depend on . Without using the WLLN...
v Problem 5 Let Xi, і ї, , n, n-256, be i.i.d. Pois(1)-random variables, and Sn- il Xi. a) Using Chebychev's inequality, estimate the probability that P(Sn > 2E S]).
Exercise 6.14 Let y be distributed Bernoulli P(y = 1) unknown 0<p<1 p and P(y = 0) = 1-p f or Some (a) Show that p E( (b) Write down the natural moment estimator p of . (c) Find var (p) (d) Find the asymptotic distribution of vn (-p) as no. as n> OO.
Let X ~ Geomeric(p). Using Chebyshev's inequality find an upper bound for P(|X – E[X]] >b).
Let X, Y be two independent exponential random variables with means 1 and 3, respectively. Find P(X> Y)
For some n > 1, let T E End(Pn) be given by T(p) = p'. Show that T is not diagonalizable.
Problem 2: Let X be a binomially distributed random variable based on n 10 trials with success probability p 0.3. a) Compute P(X 3 8), P(x-7 and PX> 6) by hand, showing your work.
b. Let U2 u~xã. Show that E(b)=n-2 for n > 2. LU
7. Let X1, X2, ... be an i.i.d. random variables. (a) Show that max(X1,... , X,n)/n >0 in probability if nP(Xn > n) -» 0. (b) Find a random variable Y satisfying nP(Y > n) ->0 and E(Y) = Oo
4. Let y1θ ~iid Uniform (0,0), for i-1, n, Assume the prior distribution for θ to , be Pareto(a, b), where p()b1 for 0> a and 0 otherwise. Find the posterior distribution of θ.
Let X, Y ~ 10,11 independently. Find P(max(X, Y} > 0.8 1 min(X, Y} = 0.5)