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19. By symmetry, the answer is (A). 25. The answer is (D). To see this, note that each door either belongs to exactly two rooms (as it leads belongs to a room and leads to another room), or belongs to only one room (as it leads outside the house). We are told N is the number of doors that lead...

$\noindent Another way to derive the result is as follows. Let $S = \sum _{k=1}^{n}\sin (kx)$. Multiply both sides by $\sin \frac{x}{2}$: $\left(\sin \frac{x}{2}\right)S = \sum_{k=1}^{n} \sin (kx)\sin \frac{x}{2}$. Now, from products to sums, we have $\sin (kx)\sin \frac{x}{2} =...

Just replace any e^{ix} you see with cis(x) (they are the same thing) and it becomes HSC-friendly.

Yeah it's negative binomial.

Basically what we do when we expand polynomial products (or products of power series) like this, to get the coefficient of x^{k}, we'll combine terms that'll give an exponent of k, which becomes like adding, due to index laws. E.g. To get the coefficient of x^{5} in...

You can read up more about PGF's and their properties here: https://en.wikipedia.org/wiki/Probability-generating_function .

They basically used a probability generating function (PGF) (except not normalised, but that doesn't matter much, just divide through by a 6 for each PGF if you want, which is what they do in the end to get the probability). Recall that the PGF of a sum of independent random variables is the...

Need to know the prefixes. k means 'kilo' (x10^3). A x10^{-9} would be nano.

Re: University Statistics Discussion Marathon $\noindent (It is clear we should have $r\geq 2$ for optimality.) One way to do it would be as follows. Let $D_{k}$ be the result of the $k$-th die roll ($k=1,2,3,\ldots$). Let $T_{r}$ be the first $n$ such that $D_{n}$ is either $1$ or $\geq r$ ($r...

Re: University Statistics Discussion Marathon Another one you can do similarly: $\noindent Show that if $X$ and $Y$ are independent geometric random variables, then $Z := \min(X, Y)$ is also geometric, and find the parameter of $Z$ in terms of those of $X$ and $Y$.$

Re: University Statistics Discussion Marathon The probabilistic interpretation: X ~ Bin(n, p) counts the no. of sucesses in n independent Bernoulli trials (with success parameter p) and Y ~ Bin(m, p) counts the no. of successes in m independent Bernoulli trials. So if X and Y are...

I think it wouldn't make sense or any use to define log to base 1. Its graph would essentially be the vertical line x = 1. Staying in the world of real numbers, 1^{t} = 1 for all real t (so raising 1 to any positive power can only yield 1 as an output). For x > 0, log1 x (which is the number 1...

Re: MATH1231/1241/1251 SOS Thread $\noindent Need to define complex powers and logarithms to make sense of something like $i^i$. I'll be slightly brief now (search up things like complex logarithm and powers online for more info). We generally define (for $z$ and $c$ complex and $z \neq 0$)...

Re: HSC 2017 Maths (Advanced) Marathon $\noindent Given $n$ real numbers $x_{1},x_{2},\ldots, x_{n}$ ($n$ a positive integer), a \textsl{weighted average} of these is a combination of them of the form $\sum_{i=1}^{n}\alpha_{i}x_{i}$, where the $\alpha_{i}$ are non-negative numbers that sum to...

It is easy to show that function f(x,y) = (xy)/(x+y) (x, y > 0) is not convex (e.g. along the line y = 1, the function is x/(x+1) (x > 0), which is not convex). To get the inequality direction you wrote, we'd want f(x,y) to be concave, so you should try investigating concavity rather...

If f is convex, the inequality should be the other way round.

$\noindent (We don't actually need $p$ to be prime, any positive integer at least $2$ will do.) If we sum from $1$ to $p$, then we are summing the real parts of all the $p$-th roots of unity, which gives 0, because the sum of roots of the polynomial $f(z) = z^{p} -1$ is 0. Since the final term...

Suppose a and b are congruent mod m, then we can write a = b + km for some integer k. Now, let d1 = gcd(a,m), and let d2 = gcd(b,m). Since d2 | b and d2 | m, we have d2 | (b + km), so d2 | a. So d2 divides both a and m, so d2 ≤ d1. A similar argument shows d1 ≤ d2 (since b = a + Km, where K =...

It suffices to find the expected payment in refunds for 1 printer and just multiply this by 100 (due to linearity of expectation) To find this for 1 printer, calculate the following probabilities: a := Probablity of failure in the first year (i.e. Pr(X < 1)) b := Probability of failure in...

$\noindent Usually we have $L$ as the independent variable and we measure $T$ for varying values of $L$. So we should obtain a plot of $T^{2}$ vs. $L$ to which we can fit a line of best fit. The slope of this line should theoretically be $\frac{4\pi^{2}}{g}$. So if we have the slope of the LOBF...