Problem 10(20). Let x and y be vectors in R". Prove that |x"y| < ||x|||y- No...
1. Let x, a € R. Prove that if a <a, then -a < x <a.
Problem 10. Let f,g: [a,b] -R be Riemann integrable functions such that f(x) < g(x) for all x E [a,b]. Prove that g(x)
(7 pts.) Let f(x, y, z) = "y and let R be the region {(x, y, z) |2 < x < 4,0 Sy < 3,15 zse}. 2 Evaluate | $180,0,.2) av. R
2. Let R be the region R = {(X,Y)|X2 + y2 < 2} and let (X,Y) be a pair of random variables that is distributed uniformly on this region. That is fx,y(x, y) is constant in this region and 0 elsewhere. State the sample space and find the probability that the random variable x2 + y2 is less than 1, P[X2 +Y? < 1].
Let X be a continuous random variable. Prove that: P(21-; < X < xạ) = 1 - a.
Let z=5 where x, y, z E R. Prove that z? +z2+z?>
Let X N(1,3) and Y~ N(2,4), where X and Y are independent 1. P(X <4)-? P(Y < 1) =? 4、 5, P(Y < 6) =? 7, P(X + Y < 4) =?
Let T be a bounded subset of R and let S CT. Prove that supS < supT.
an (4) Let F be ordered field. Prove that the statement Vo: ZX\x6F*XWye FtX &<y)->(<y') is true (hist: Factor Y-X out of yº-x")
Let x[n] and y[n] be periodic signals with common period N, and let z[n] = { x[r]y[n – r) r=<N> be their period convolution. Let z[n] = sin(7") and y[n] = { . 0 <n<3 4 <n <7 Asns? be two signals that are periodic with period 8. Find the Fourier series representation for the periodic convolution of these signals.