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Re: Statistics In this case X and Y are not independent, because the product of the marginal densities is not the same as the joint density.

Re: Statistics Almost sure convergence is stronger than convergence in probability because the former implies the latter, but the latter generally does not imply the former. You can either try proving this as an exercise or finding a proof online, there should be one on Wikipedia even, I...

Re: MATH1131 help thread $\noindent He-Mann multiplied top and bottom by the conjugate of the denominator, because this makes the denominator real and thus makes it easy to find the real part after simplifying the numerator. (When the denominator isn't real, we can't just see what the real...

Re: MATH1131 help thread $\noindent Note $z \neq -1$, otherwise we divide by $0$. Assume $|z| = 1 \iff z \bar{z} = 1$ and $z \neq -1$. Try and use $\mathrm{Re}\left(w\right) = \frac{1}{2}\left(w + \overline{w}\right)$ with $w = \frac{1-z}{1+z}$.$

$\noindent Let the desired polynomial be $P(x)$. Since $P$ is odd, we must have that $2$ is also a triple root and $-1$ is also a root. So $P(x) = (x+2)^{3}(x-2)^{3}(x+1)(x-1)Q(x) = \left(x^{2} -4\right)^{3}\left(x^{2} -1\right)Q(x)$ for some polynomial $Q(x)$ (which must be monic for $P(x)$ to...

Re: University Statistics Discussion Marathon $\noindent An example of such a dataset: $n = 2$, with $x_{1} = \frac{2-\sqrt{2}}{2}, x_{2} = \frac{2 + \sqrt{2}}{2}$.$

Re: University Statistics Discussion Marathon $\noindent Use the sample variance formula: $S^{2} = \frac{1}{n-1}\sum_{i = 1}^{n}\left(x_{i} - \overline{x}\right)^{2}$, where $S^{2}$ is the sample variance and $\overline{x}$ is the sample mean. Note that if the (sample) standard deviation...

Re: University Statistics Discussion Marathon Third option

In what sense do you feel that "2U students are not being adequately prepared for assessment tasks and are given a bit of a '2nd class citizen' treatment"? Do you mean the teachers aren't bothering to teach 2U students as well as they are 3U students?

You can get in via graduate entry without needing a 99.95 ATAR. The 99.95 requirement is just for those who'll get a spot from high-school leavers. So the person with 96 ATAR at USyd Medicine would probably be from graduate entry.

Re: MATH1131 help thread Correct!

No

Re: MATH1131 help thread $\noindent Recall that $\frac{\mathrm{d}}{\mathrm{d}t}\left(\sin t \right) = \cos t$ and $\frac{\mathrm{d}}{\mathrm{d}t}\left(\cos t \right) = -\sin t$. Also make sure to recall the chain rule. $

Re: Statistics $\noindent We want the (expected) \emph{area}, whilst $X$ is the \emph{diameter}, so the area random variable $A$ is $A =\frac{\pi}{4}X^{2}$. (Remember the formula area of a circle given its diameter.) Hence to find the expected area, we need to find the expected value of...

Re: Statistics $\noindent [Aside: the distribution with that pmf is the \emph{geometric distribution on } $\mathbb{N}:= \left\{ 0,1,2,\ldots \right\}$ with parameter $p$.]$ $\noindent It is really only of interest to find the CDF for non-negative integer values. For each $k\in \mathbb{N}$...

Re: Statistics $\noindent What do you mean a form with $x$? You don't need to express it as a formula like $f(x) = \frac{\binom{4}{x}\binom{2}{2-x}}{\binom{6}{x}}$ or something (this is the formula for it though if you're curious) if that's what you mean. Writing it as I wrote it is fine...

Re: Statistics $\noindent Those answers aren't right (assuming sampling without replacement). Note your pmf values don't sum to $1$, which they should. This'll just be a hypergeometric distribution, but you don't really need to know that here. We have, letting $f(k) \equiv f_{X}(k) =...

Re: International Baccalaureate Maths Marathon $\noindent Let's recall some discriminant theory. Let $Q(x) = ax^{2} + bx + c$ be a real quadratic (i.e. the coefficients $a,b,c$ are real numbers) and $a \neq 0$. Let $\Delta = b^{2} - 4ac$ be the discriminant. Then we have$ $$\bullet \, \Delta...

Re: International Baccalaureate Maths Marathon Why do you think they don't know how to teach the subject properly there? Is it something their worked answers said? The reason we can set the discriminant to be less than or equal to 0 is that we are told the quadratic is non-negative for all...

Re: International Baccalaureate Maths Marathon $\noindent The question is correct. The discriminant inequality should be the other way round, i.e. $\Delta \leq 0$ if the quadratic is non-negative for all real $x$.$