Recent content by Yagoda

Thanks for providing a nice, intuitive approach. I don't think I would be clever enough to come up with that under the time pressure of an exam. :eek: That numerical calculation is still a mystery to me.

Homework Statement Toss a fair coin independently 100 times. Let X > 0 be the number of times the coin must be tossed until the number of observed heads equals the number of observed tails. (And let X=100 if this never happens). Find the probability that X=8. Homework Equations The...

Homework Statement I want to show that if I have a consistent sequence of estimators W_n for \theta, i.e. \lim_{n \rightarrow \infty} P(|W_n - \theta| < \epsilon) = 1, then U_n = a_nW_n + b_n is also a consistent sequence of estimators for \theta where \lim_{n \rightarrow \infty}a_n = 1 and...

I did try one, but I was having trouble keeping track of all the relevant numbers: possible flips, number of heads, corresponding probabilities that I was hoping there might be a clearer way. I might have to give it another crack...

Homework Statement The following experiment involves a single coin with probability p of heads on anyone flip, where 0 < p < 1. Step 1: Flip the coin. Let X = 1 if heads, 0 otherwise. Step 2: Flip the coin (X + 1) times. Let Y = the number of heads obtained in this step. Step 3: Flip the...

Homework Statement Let h(u,v) = f(u+v, u-v). Show that f_{xx} - f_{yy} = h_{uv} and f_{xx} + f_{yy} = \frac12(h_{uu}+h_{vv}) . Homework Equations The Attempt at a Solution I'm always confused on how to tackle these types of questions because there isn't an actual function to...

After fiddling a little bit with the inequalities, it seems like in the general case it would be helpful to show that \frac{x_1 + x_2 + \dots + x_n}{\sqrt{x_1^2 + \dots + x_n^2}} < \sqrt{n}. Would this be a proper approach? And if so, is there some glaringly obvious fact I'm missing that would...

If X = (1,1,1) then |\textbf{Ax}| = a \sqrt{2}\sqrt{3}, but I guess I am missing what happens if X is a different vector, say (1,2,3). Then |\textbf{Ax}| = a \sqrt{2}\sqrt{14}, which isn't less than a \sqrt{2}\sqrt{3}. I think in this case \textbf{||A||}= \frac{a\sqrt{72}}{\sqrt{14}} so in this...

Homework Statement Let \textbf{A} be an m x n matrix and \lambda = \max\{ |a_{ij}| : 1 \leq i \leq m, 1 \leq j \leq n \}. Show that the norm of the matrix ||\textbf{A}|| \leq \lambda \sqrt{mn}. Homework Equations The definition I have of the norm is that ||\textbf{A}|| is the smallest...

Homework Statement Let h(u,v) = f(a(u,v), b(u,v)), where a_u = b_v and a_v = -b_u. Show that h_{uu} + h_{vv} = (f_{xx} + f_{yy}) (a^2_u + a^2_v). Homework Equations The Attempt at a Solution I suppose my first question is where the x's and y's come from. (I thought at first it...

Thanks for the explanation. I feel like in the text and in class we've barely scratched the surface of what is needed for a rigorous proof, but this helps. I'll see where I get. The following question after this one is to show that if f(x) = g(x) almost everywhere then their Riemann integrals...

I am trying to show Riemann integrability. We haven't covered the Lebesgue integral, just the criterion. Yes, I do see now that being non-zero only on a null set, does not imply boundedness. If the Riemann integral does exist, the function must be bounded though, correct? Well, the null...

Homework Statement Let h(x) = 0 for all x in [a,b] except for on a set of measure zero. Show that if \int_a^b h(x) \, dx exists it equals 0. We are given the hint that a subset of a set of measure zero also has measure 0. Homework Equations We've discussed the Lebesgue integrability...

Since X_1 and X_2 are both smaller than 1, then their product will be smaller than X_1 alone, but I'm not seeing how this helps me find the pdf of Y. Are the limits 0 < y < z not correct? It seems that this needs to be true in order for Y/Z to be less than 1? Now that I'm looking at it...

Homework Statement Two RVs X1 and X2 are continuous and have joint pdf f_{X_1,X_2}(x_1, x_2) = \begin{cases} x_1+x_2 &\mbox{for } 0 < x_1 < 1; 0 < x_2 < 1 \\ 0 & \mbox{ } \text{otherwise}. \end{cases} Find the pdf of Y = X_1X_2.Homework Equations I'm using the transformation "shortcut' that...