Recent content by Karnage1993

Suppose I have observed ##Z = 3##, where ##Z = X + Y##, where ##X \sim N(0,9), Y \sim N(0,4)##. How would I find the most probable value of ##X## that would have given me ##Z = 3##? My attempt at a solution: I was given that ##X## and ##Y## are independent, so that means ##Z \sim N(0+0, 9+4) =...

I was under the impression that ##\textrm{rank}(A) = \textrm{rank}(A^TA)## is only true if ##A## is symmetric, but it appears you are right, and Wikipedia confirms it. It is indeed true in general for any ##A##, so I guess I misread it somewhere. Thanks for the help!

Isn't ##\textrm{rank}(A) = \textrm{rank}(A^TA)## only true if ##A## is symmetric? Also, I forgot to include that ##A## is not necessarily a square matrix. Let's have ##A## be an ##n## x ##k## matrix. Does your conclusion still follow with these new conditions?

Say I have a matrix ##A## that has linearly independent columns. Then clearly ##A^T## has lin. indep. rows. So what can we say about ##A^TA##? Specifically, is there anything we can say about the rows/columns of ##A^TA##? I'm thinking there has to be some sort of relation but I don't know what...

Homework Statement Suppose ##X,Y## are random variables and ##\varepsilon = Y - E(Y|X)##. Show that ##Cov(\varepsilon , E(Y|X)) = 0##. Homework Equations ##E(\varepsilon) = E(\varepsilon | X) = 0## ##E(Y^2) < \infty## The Attempt at a Solution ##Cov(\varepsilon , E(Y|X)) =...

I'm working on a problem that wants me to show that $$Cov(X,Y) = 0$$ and I am up to the point where I simplified it down to $$Cov(X,Y) = E(XY)$$. In other words, $$E(X)E(Y) = 0$$ to make the above true. My question is, what can we conclude if we have that the covariance of two random variables...

Homework Statement Evaluate ##\int_{\gamma} F \ ds## where ##\gamma## is a parametrization of the curve of intersection of the surface ##z = x^4 + y^6## with the ellipsoid ##x^2 + 4y^2 + 9z^2 = 36## oriented in the counterclockwise direction when viewed from above. Homework Equations...

Yes, I believe so. (That is for ##0 \le r \le a, 0 \le \theta \le 2\pi##, for some finite ##a##, right?) The process to get to it is quite lengthy, but I can do it.

Homework Statement Find the steady state temperature ##U(r, \theta)## in one-eighth of a circular ring shown below: Homework Equations The Attempt at a Solution I start by assuming a solution of the form ##u(r,\theta) = R(r)\Theta(\theta)##. I also note that ##u(r,\theta)##...

For it to be outward pointing, would the z-component have to be negative, ie, (0,0,-1)?

Ok, so what I think you're saying is I have to do: ##0 - \int_{plane \ z = 0} F \ dS## which will get me only the surface integral of the graph of z?

Homework Statement Let ##\mathit{F}(x,y,z) = (e^y\cos z, \sqrt{x^3 + 1}\sin z, x^2 + y^2 + 3)## and let ##S## be the graph of ##z = (1-x^2-y^2)e^{(1-x^2-3y^2)}## for ##z \ge 0##, oriented by the upward unit normal. Evaluate ##\int_{S} \mathit{F} \ dS##. (Hint: Close up this surface and use the...

Okay, so suppose one piece of the surface is when ##x = cy##. How would I go about finding the limits for ##c,y##?

I don't quite get why you set ##z = c^2 \ge 0##. For example, if ##x=0,y=0## and ##z## negative, that also satisfies the equation, right?

Homework Statement Let S be the self-intersecting rectangle in ##\mathbb{R}^3## given by the implicit equation ##x^2−y^2z = 0##. Find a parametrization for S.Homework Equations The Attempt at a Solution This is my first encounter with a surface like this. The first thing that came to my mind...