I. Let {X n\ be a sequence of random variables wit h E(X,-? for n- 7n...
Problem 2. Let f be a self-map on a set X. For x,y e X define x ~ y if and only if f"(x) = f(y) for some integers n, m > 0. Show that ~ is an equivalence relation.
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
Let X be an exponential random variable such that P(X<26) = P(X > 26). Calculate E[X|X > 28]. Answer: CHECK
PROB 4 Let Xi and X2 be independent exponential random variables each having parameter 1 i.e. fx(x) = le-21, x > 0, (i = 1,2). Let Y1 = X1 + X2 and Y2 = ex. Find the joint p.d.f of Yi and Y2.
Let X1, ..., Xn be a random sample from the distribution 1 f(x; 01, 02) e-(2–01)/02 x > 01, - < 01 <0, 02 > 0. 7 02 Find the method of moments estimators (MMEs) of 04 and 02.
check if e-1/4/ f(x) if x > 0 if x < 0 is differentiable at 0.
Consider the pair of random variables (X,Y). Suppose that marginally X ~ Binomial(2, ) and Y ~ Binomial(2, 3). If P(X > Y) = 0 and P(X = 0, Y = 2) = 16, then P(X = 1, Y = 1) equals
Let us consider the biased random walk S,-X1 +X2+···+X,, with S: is a sequence of independent randon variables with P(X,--1)-g,P(X.-1) p+q=1 and Mt Show that M.-Sn-b- calculate Elr), where τ = inf{n : S" = a or-6) with a, b > 0. 0. where Xi, X2 p. where g)n is a martingale. Use this martingale to
4.3. Let X and Y be independent random variables uniformly distributed over the interval [θ-, θ + ] for some fixed θ. Show that W X-Y has a distribution that is independent of θ with density function for lwl > 1.
please explain each step 5.21 Let X and Y be independent random variables with fae-ax, x>0 fx(x) = 10. otherwise and Be-Bt, x>0 fr(y) = 10 otherwise where a and B are assumed to be positive constants. Find the PDF of X + Y and treat the special case a = B separately.