Σ.ix Problem 8. I et Xi,Xy ,x, be independert (0, 1) randoua variables. Let rn.- for...
Problem 8. Let X1, X2, , Xn be independent ฆ(0,1) random variables. Let m,-1 for k 1,2,3. Are there numbers mi,m2, m3 such that n.y rn1 m1 a.S n3 m3 holds? If so, calculate these numbers.
Problem 3: Let xi be given n mutually orthogonal vectors in Rn, and 20 є Rn be also given. Find: (a) the distance di from Zo to Hi-{x E Rn : XTXǐ (b) the distance sk from ro to n1Hi, 1 <k< n (c) the distance mk from a'0 to ngk+1H,, 1-K n (d) calculate sk + mk 0)
Problem 3: Let xi be given n mutually orthogonal vectors in Rn, and 20 є Rn be also given. Find: (a)...
3, Let X, X2,X, be independent random variables such that Xi~N(?) a. Find the distribution of Y= a1X1+azX2+ i.(Hint: The MGF of Xi is Mx, (t) et+(1/2)t) + anXn +b where a, 0 for at least one b. Assume = 2 =n= u and of- a= (X-)/(0/n) ? Explain. a. What is the distribution of The Sqve o tubat num c. What is the distribution of [(X-4)/(0/Vm? Explain.
Problem 9. Let Xi, X2,... , Xn be independent 2/ (0,1) random variables. Set F(t) Is there a matrix M such that holds with independent standard normal random variables Z1, Z2, Z3? If so, calculate M.
Let xi be independent. E(xi)=0. Var(xi)= sigma ^2
Cov(x,y) = E(XY) - ExEy
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1. Let ω be a k-form in Rn , Π 〉 k. If k is odd, show that ωΛω 0 4. L et ω be a k-form and let ) be a /-form. Fin d d(da) Λ η_ωΛ đ7) .
1. Let ω be a k-form in Rn , Π 〉 k. If k is odd, show that ωΛω 0 4. L et ω be a k-form and let ) be a /-form. Fin d d(da) Λ η_ωΛ đ7) .
C2.3 Let X and Y be random variables with finite variance, so that EX2o0 (i) Show that E(X) - (EX) E(X - EX)2, and hence that the variance of (ii) By considering (|XI Y)2, or otherwise, show that XY has finite expecta- (iii) Let q(t) = E(X + tY)2. Show that q(t)2 0, and by considering the roots of and EY2 < oo. X is always non-negative. tion the equation q(t) 0, deduce that
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.
REDIT (10 pts). Suppose X = (Xi,Xy, ,x,000) are random variables taking values S Xi S 1 for all 1). Design a hypothesis test that tests the null hypothesis that 1, X', ...X,oo are iid (independent and identically-distributed) and uniform on [0, 1] at significance level α. (Recall that the best way to do this is to i) choose a statistic S(X) that you can compute the tion of, assuming the null hypothesis is true, and ii) use that to...
5. Let Xi, , X, (n 3) be iid Bernoulli random variables with parameter θ with 0<θ<1. Let T = Σ_iXi and 0 otherwiase. (a) Derive Eo[6(X,, X.)]. (b) Derive Ee16(X, . . . , Xn)IT = t], for t = 0, i, . . . , n.