Recent content by rwinston

Ah, yes...sorry, a bit obvious really when you think about it: \sum_{i=1}^{N}x_i^2 -N\bar{x}^2 = \sum_{i=1}^{N}x_i^2 -\sum_{i=1}^N\bar{x}^2 \sum_{i=1}^{N}\left( x_i - \bar{x}\right)^2 And the other bit: \sum_{i=1}^{N}x_iy_i - N\bar{x}\bar{y} =...

Hi Sorry this may be an obvious one...can anyone help me with getting from the first to the second equation below? I'm particularly stuck with manipulating the terms inside the summations formulas. I can derive to here: \sum_{i=1}^{N}x_iy_i - N\bar{x}\bar{y} - \left(...

Hi I may be missing something obvious here, but I am reading a paper on random sampling, and in one of the proofs, two consecutive steps run like this: = \sum_{d=0}^m \left[ {{m}\choose{d}} \left(\frac{1}{n}\right)^d \left(\frac{n-1}{n}\right)^{m-d} \right] \left( \frac{m-d}{m}\right) =...

Here is a probability question I came across recently: If the chance of being hit by a car while crossing the street is 0.1, then what are the chances of being hit by a car after crossing the street 10 times? Obviously the probability can't be 100% (the sum of the individual...

Hmm. Ok if u = 1 + x^2 du = 2x dx \int x \sqrt{1+x^2}dx = \frac{1}{2}\int \sqrt{1+x^2}2xdx =\frac{1}{2} \int\sqrt{u}du =\frac{1}{3} (1+x^2)^{\frac{3}{2}} But if I take u^2=1+x^2 u = \sqrt{1+x^2} du = \frac{x}{\sqrt{1+x^2}}dx \int{x\sqrt{1+x^2}dx} =...

Hi I am trying to integrate x \sqrt{1+x^2}dx by parts...but it seems to involve trigonometric functions - is it possible to solve this integral without using trig functions? Thx

Thanks Leonardo! That makes sense now :-)

Thanks! Thats exactly what I need!

Hi does anybody know how to create a labelled matrix in LaTeX? By which I mean something like this: a b a [ 1 0 ] b [ 0 1 ] Where the inner elements are within the braces, but there are tabular label elements for the rows and columns (a and b). Cant get it too look right! Cheers

Actually I think I was making this more complicated than it needed to be. Z(t;T) = exp^{-\int_{t}^{T}{r(\tau) d\tau}} -\log{Z(t;T)}=\int_{t}^{T}{r(\tau) d\tau} -\frac{\partial}{\partial{T}}=r(T)-r(t) r(t)=0, so: r(T) = -\frac{\partial}{\partial{T}}\log{Z(t;T)} Im still not sure why...

Thanks guys - yes, sorry, I had a typo in the integral - it should have been T as the upper limit of integration.

Hi I have a question about rearranging the following equation (I saw this in a finance book): If we rearrange and differentiate Z(t;T) = e^{-\int_{t}^{\tau}r(\tau)d\tau} We get r(T) = -\frac{\partial}{\partial{T}}(\log{Z(t;T)}) My question is: how do we differentiate...

Thanks Ramsey. I didnt think there would be a straightforward answer to this one.

Hi Following on from my palindromic number question, if I am calculating a set of palindromic numbers generated as the product of two three-digit numbers, I could generate the entire product set and then test each number for the palindrome property. the naive approach would be a loop like...

Hi all I was looking q # 4 on ProjectEuler.net, and it is a problem to find the largest palindrome constructed from the product of two three-digit integers. Now, the answer (spoilers, if you don't want to know ) is 906609. However, I got a different answer, but possibly my logic is incorrect...