Recent content by SithsNGiggles

Yes thank you I noticed that too as I was writing my answer. Thanks for the help!

Hey everyone, I think I figured out the justification part, sans induction. What I did was set up a diagram, like so: ##\_~\_\cdots\_~\_##, where there are ##n## slots for one of the three letters. Then I considered the possible outcomes you can get if you were to add an additional letter to...

Yeah it looks like I'll have to settle with that, at least until I can get a hint from my prof. Thanks to everyone that replied!

The previous question asked me to compute the first four terms. I believe I'm treating the empty string as a string that doesn't end in ##b##, so ##f_0(0)=1##. I'll try to confirm with my professor. Yeah, I get what the relations represent, I just don't understand what it means to "justify"...

Hi haruspex. I think I left out some details regarding ##f_0## and ##f_1##. The former counts the number of strings of length ##n## that do not end in ##b##, and the latter counts those that DO end in ##b##. So, for example, when n=2, we can have ##\{aa,ac,bc,ca,cc\}##, so ##f_0(2)=5##. I've...

Homework Statement I'm given a recursive sequence with the following initial terms: ##\begin{matrix} f_0(0)=1&&&f_1(0)=0\\ f_0(1)=2&&&f_1(1)=1 \end{matrix}## Now, I'm asked to justify that we have the following recursive relations: ##\begin{cases} f_0(n)=2f_0(n-1)+f_1(n-1)\\...

Oh right, I forgot that ##W_n=\frac{1}{n}\sum\cdots##. Thanks! I meant to say ##E(W_n)=1##.

Judging by ##W_n##'s distribution, that would be ##n##. Does that mean ##c=n##, and if so, is that always the case (that the mean is the constant I'm supposed to find)?

Homework Statement Let ##Y_1,...Y_n## be independent standard normal random variables. What is the distribution of ##\displaystyle\sum_{i=1}^n{Y_i}^2## ? Let ##W_n=\displaystyle\frac{1}{n}\sum_{i=1}^n {Y_i}^2##. Does ##W_n\xrightarrow{p}c## for some constant ##c##? If so, what is the...

Homework Statement I came across this integral recently while tutoring: ##\displaystyle \frac{1}{5} \int \frac{-x^3+2x^2-3x+4}{x^4-x^3+x^2-x+1}~dx## Homework Equations The Attempt at a Solution I'm not sure how to approach this. At first I suspected partial fraction decomposition...

Thanks, I noticed my mistake a few moments ago. For this particular operation (a*b = a^3 + b^4), would it suffice to say something along the lines of: (as per Bacle2's observation) If there's anything wrong with the syntax, please let me know. Thanks

Hey, that's a pretty neat observation! It got me wondering: perhaps if the powers of a and b were odd, then the operation might be injective? I'll see if that gets me anywhere. Thanks again to everyone else as well for the replies.

@DaxInvader, you get these equations by matching up the vector terms containing ##e^x## and ##xe^x## on both sides.

##n=1## and ##m>1## come to mind. Beyond that, I'm not sure. I'll put some more thought into it.

@HallsofIvy, how do I use that to show a = c and b = d? I see if they are equal, then I get the identity 0 = 0, but I don't think that proves anything. Could it be that ##(a-c)(a^2+ac+c^2)\not=(d-b)(d^3+bd^2+b^2d+b^3)## if ##a\not=c## and ##b\not=d##? Thanks