Recent content by LoadedAnvils

What is the difference between integrating over the angles and integrating over the surface parameterised by these angles? I shouldn't have put S as a function. If I do an integral on each side, I get the desired result if it is an integral over the angles.

Homework Statement Suppose I have a relation S(\theta, \phi) = U \frac{c}{2} \cos{^{2} \, \theta} and I want to integrate over \phi from 0 to 2 \pi and \theta from 0 to \frac{\pi}{2} . How do I do this double integral? Do I just do it normally (without any transformation), or do I...

The textbook defines E_{\theta} \left( r(X) \right) = \int r(x) f(x; \theta) dx. What I did is just evaluated the expectation of \hat{\lambda}.

Thanks. However, I still want to know if I calculated this correctly (as I will be doing the same for calculating the standard error and MSE).

Let X_{1}, \ldots, X_{n} \; \mathtt{\sim} \; \textrm{Poisson} (\lambda) and let \hat{\lambda} = n^{-1} \sum_{i = 1}^{n} X_{i}. The bias of \hat{\lambda} is \mathbb{E}_{\lambda} (\hat{\lambda}) - \lambda. Since X_{i} \; \mathtt{\sim} \; \textrm{Poisson} (\lambda), and all X_{i} are IID...

Oh! I get it now! Thank you so much!

What do you mean by limiting property?

I'm aware of that, and I know how to prove it for the case of X \geq 0, but I'm confused about the case of the entire real line. Also, what do you mean by \overline{x}?

If X is a continuous random variable and E(X) exists, does the limit as x→∞ of x[1 - F(x)] = 0? I encountered this, but so far I have neither been able to prove this, nor find a counterexample. I have tried the mathematical definition of the limit, l'Hopital's rule, integration by parts, a...

I finally see. I will try to learn measure theory, but for now, thank you.

Y is discrete, so f_Y (y) = P(Y = y) = P(r(X) = y) r(X) = y is equivalent to X ∈ A_y X ∈ A_y → {ω: X(ω) ∈ A_y} = U(x ∈ A_y) of {ω: X(ω) = x} Since each {ω: X(ω) = x} is disjoint for distinct x, P(X ∈ A_y) = Sum of P(X = x) for x ∈ A_y Can you show me at which step I am wrong exactly...

Can you explain why that step is not justified? I rewrote a proof that they are equivalent and it seems to hold for both continuous and discrete random variables. Here it is as an attachment. Sorry, I'm still very new to probability and I'm trying to understand measure theory.

Link to theorem: http://en.wikipedia.org/wiki/Law_of_the_unconscious_statistician Suppose Y is a discrete random variable related to X, a continuous random variable by some function r (so Y = r(X) ). Let A be the following set: A_y = {x ∈ R ; r(x) = y}. Since Y is discrete, f_Y (y) = P(Y = y)...

I understand energy-momentum tensor with contravariant indices, where I think I get T^{αβ}, but how do I derive the same result for T_{αβ}? Why are the contravariant vectors simply changed to covariant ones, and why does it work in Einstein's equation?

For the first one, remember that the distance depends on the time it takes for light to reach the other object. c is the speed of light in a vacuum, but through a medium the speed of light is less. Since visually distance depends on time, you can find out the distance. Second, the angle of...