Recent content by GJA

Hi Dhamnekar Winod, The formula you posted is for a real-valued function, but the function $f$ you provided is a vector field. You will need to use the formula for a surface integral of a vector field: $$\iint\limits_{R}f(x(u,v), y(u,v), z(u,v))\cdot \left(\frac{\partial\mathbf{r}}{\partial...

Hi Peter, The reason the author is doing things this way is because the outer measure is defined as an infimum. Recall that the infimum of a set of real numbers is the greatest lower bound of that set of real numbers. According to Definition 2.2, if for an arbitrarily fixed $\varepsilon >0$...

Hi Dan, I like the symbolic argument above. Here is something I scratched out in an attempt to avoid the $F^{-1}(n)$ term: By definition of the inverse, $$\frac{1}{\phi(\alpha E)}\phi(\alpha E) F(n) = F(n)$$ From (1) we get $$\left[\frac{1}{\phi(\alpha E)}\alpha^{-n}\phi(E)\alpha^{n}\right]...

Hi kimchuu, Question 1 I agree with you that this histogram is difficult to read. The bar for 31-40 is indeed above 4 slightly. The thing to keep in mind here is that the vertical axis represents number of players, which can't be a decimal value like 4.5. Unfortunately, this graph is not the...

I'm usually not one for replying to answered threads, especially in light of the expert help already provided by topsquark and County Boy. That said, I'm detailing another solution outline here because the method on which it's based, Duhamel's Principle, is one of my favorites. What I like about...

Hi Peter Question 1 I will prove the existence of $p_{j}$; the argument for $q_{j}$ is similar. Since the integers are not bounded above, the set $\{p\in\mathbb{Z}\, : \, na_{j}\leq p\}$ is non-empty. By the well-ordering principle, $\{p\in\mathbb{Z}\, : \, na_{j}\leq p\}$ has a least...

Hi Dhamnekar Winod, Thanks for posting this question; glad to hear you were able to sort it out. I'm adding a solution outline below in case there are MHB community members who are interested in learning about the "stars and bars" combinatorial principle that you've correctly applied (after...

Hi Peter Proof looks good. Two small comments: It is true that since both $f$ and $g$ are bounded with compact support, it is possible to select a single rectangle $B$ for which $f(x) = g(x) = 0$ for $x\notin B$. I think to improve the rigor of your proof, it is worth saying something like...

Hi Peter Nice job once again looking to $\mathbb{R}^{2}$. Your pictures are also very well done and illustrate the concepts really well. You asked a number of questions in this post and, as far as I can tell, your answers and intuition are correct. A few additional notes are: "presumably...

Hi Peter, Great job looking to $\mathbb{R}^{2}$ to gain an intuition for what's going on. Also really nice that you've posited a number of possible interpretations of what the author has written. You've done a great job outlining testable criteria, so let's do just that. Suppose the author...

Hi mathmari, Good work so far. Here are a few ideas to keep things moving. (a) You're correct that $\max(X_1,\ldots , X_n) \leq y$ iff $X_i\leq y$ for all $1\leq i\leq n$. It is also true in this case that $F(y) = \left (P_{\theta}[X_1\leq y]\right )^n$, but this does not follow just because...

Here is a plot of the likelihood function for part (c), where $n=10$ and $\sum_{i}x_{i} = 40$. As you can see, it's maximized for $\theta=0.25$, as it should be from your derivation in part (b).

You're correct on everything; great job!

Exactly, nice job! For part (b) you're trying to determine the value for $\theta$ that maximizes $L_{x}(\theta)$ for the given data $x_{i}$; i.e., you're thinking of the $x_{i}$ as fixed so that $L_{x}(\theta)$ is being considered as a function of $\theta$. Using the fact that...

Hi mathmari, Nice job so far. Here are a few ideas to keep things moving along. (a) The product should be for $1\leq i\leq n$. Using this fact and the formula for $p_{\theta}(x_{i})$, we can take your work a step further to obtain a closed-form expression for the likelihood. (b) You will need...