Let z=5 where x, y, z E R. Prove that z? +z2+z?>
Given f(x,y) = 2 ; 0 <X<y< 1 a. Prove that f(x,y) is a joint pdf b. Find the correlation coefficient of X and Y
(a) Show that E ε4 and use it to deduce that α > 0. (b) Show that εσ σ and use it to deduce that ơ--0: (c) Use Part (b) to show that αβ -1 and α + β-_1. (d) Use Part (c) to find a quadratic polynomial in z with real coefficients whose roots (e) Use Part (d) to express cos(2π/5) using radical sign (no trig functions, no deci- are α and β mals).
Let B C R" be any set. Define C = {x € R" | d(x,y) < 1 for some y E B) Show that C is open.
(6) Let a be a positive real number. Note that for all r, y R there exists a unique k E Z and a unique 0 Sr <a so that Denote (0. a), the half open interval, by Ra and define the following "addi- tion" on Rg. where r yr + ka and r e lo,a) (a) Show that (Ra. +a) is a group (b) Show that (Ri-+i ) İs isomorphic to (R, , +a) for any" > O. (Therefore,...
Prove that A = B for: A = {(x,y) e Rº : +y/<1} B = {(z,y) € RP: (71+ y)² < 1}
5. Let y E C2([0, T]; R), T > 0 satisfy y"(t) = 피t, y(0) = y'(0) = 0 e R. Use Picard-Lindelöf 1+t' to prove that a unique solution to the IVP exists for short time, as follows: (a) Let b E R2, A E M2 (R) . Show that any function g : R2 -R2.9(x) = Ax+b is Lipschitz. 1 mark (b) Transform the DE for y into a(t) Az(t) +b(t) for a suitable z, A, b. 2...
(c) [5 points] Prove that f(r) [5 p ) = Σ (-1-rn oints Prove that f(x converges uniformly on [-c, c when 0<c<1. lenny
b-a e-ylu f(y)= e for y > 0 and L* (u ) c=constant U 1=1 i=1 Prove the likelihood for u can be expressed as: tulo: D-ring 9: 1-9 Then derive the log-likelihood for u.
9. Prove that for any positive integers a, b with a >1, b>1 and (a, b) have log,(a) is irrational. (You may use Euclid's Lemma, but not FTA.) 1, we