sophie-germain
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For part c) yes you have a set of 15 that involves all possible combinations. so each child has 15 possibilities of getting assigned candy hence $10^{15}$ is correct. yes part d) is correct.

So give -ln(X) $\sim$ exp(1) which is well established where X is $U(0,1)$, then similarly -ln(Y) $\sim exp(1)$ where Y is uniform. Then sums of exponentials are gamma(2,1) indicates that -ln(X)-ln(...

great question. so here is how to solve this. \begin{split} ET= \sum_{l=0}^{\infty}\frac{l}{c}\mu(1-\mu)^{l-1} \\ &=\frac{\mu}{c}\sum_{l=0}^{\infty}l\cdot (1-\mu)^{l-1} \\ &=...

okay, the way i understand 2. is that first consider (a,b) as an open interval, then the intersection of all closed sets [a,b] contains (a,b)... however the closure must contain the set C. thus if ...

A basis is a collection of sets, so for a mere subset, you have to show that the subset is a collection of sets, not just a single set before showing that a subset can be a basis. The second part of ...

this problem to show that your set is a topology you need to satisfy the condition of the set, and the definition of the topology. we know that if U is equal to X, then the empty set is included. if ...

I'll be the cynical response. Give it up. You have very low probability of gaining a full time research position in pure simply because online education is going to be making radical changes with how ...

Just set the equation $Ax=y$, and look at when there are solutions for $y$. Infinite solutions does not imply dependence, infinite solutions only gives a criteria of where the solution space exists ...

this function is not homomorphic and i would start looking at the range and domain of the function. is it natural numbers? reals? . assume natural numbers. then f(3)=f(5)=...f(any odd) = 20 . then ...

yes upon derivation, you should tao go to 0. Note that there is no B term. so the N' term will have a $\tau$ $\vec{B}$ term but this will go to zero.
The idea is given as follows. If the derivative of the dot product of $T\cdot T = 1$ which when taking the derivative we use the chain rule to get $T\cdot T'=0$. This just tells you that the angle ...