Clearly, the set denotes the interior of the unit circle centered at the origin. So, it contains uncountable number of points.
Its cardinality is equal to the cardinality of R.
So, the cardinality is, ¢, which is the Continuum
3 Let (X,Y) be a random vector with the pdf Se-(x+y), f(x,y) = e-(x+y) 122 (x, y) = 1 0, (x,y) E R otherwise. Find P{} <t}. In other words, find the PDF of the r.v. . Done in the class.
1. Given that {1,cos x, sin x} is a fundamental solution set for y" + y' = tanx , 0<x<5, find the particular solution using the variation of parameters method.
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 B C R" be any set. Define C = {x € R" | d(x,y) < 1 for some y E B) Show that C is open.
Let S function, f: S R, between the two sets. x < 1}. Show that S and R have the same cardinality by constructing a bijective x E R 0
1. For pdf f (r, y) = 1.22, 0 < x < 1,0 < y < 2, z +y > 1, calculate: EY) and () E (X2)
Q1 (7 points) For k e R any constant, find the general solution to xa y" + (1 – k)x y' = 0, and use it to show that when k < 0, all solutions tend to a constant as x + 2O.
With double integral find surface area when y=e", y = x, y = 4 and axis y with y <et
With the help of the Fourier series y" + y = r(x) = 2 (0<=<1) 2-2 (1<x<2) r(x+2) = r(2) Find the general solution of the differential equation
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.