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Q1 answer is cot(x). The solutions wrote tan(x) in line 3, but they had cos(x)/sin(x) before that, which is cot(x), not tan(x).

Re: Statistics Nice. Incidentally, here's what the graph of the pdf should look like...

Re: Statistics $\noindent Those bounds on the integral aren't correct. For the bounds, you will need to take cases based on whether $z \geq 0$ or $z < 0$ (refer to your diagram of the $u$-$z$ sample space). And in the end, the range of values of $Z$ will be $(-60,60)$. We could see this from...

Re: Statistics $\noindent Here's a ``by inspection'' method. We know $U = W$, so the range for $U$ will be same as that for $W$, i.e. $0 \leq u < 60$. For $Z$, we know $Z$ is just $W - V \equiv U - V$, so due to the range of values for $V$ and $U$, $Z$ can take values from $U - 60$ up to $U$...

Re: Statistics It's quite easy with the linear transforms method. The sample space in the v-w space is just a square. A square gets mapped to a parallelogram in the u-z space (in general) by a linear transformation like the one you have. (Recall that in general, an invertible linear map from...

Re: MATH2601 Linear Algebra/Group Theory Questions Yeah, that's OK for showing linear independence.

Re: Statistics Actually that function has finite limit as u -> 0, as you can show using L'Hôpital's rule for example. So the MGF does exist.

Re: Statistics $\noindent I'm not 100\% sure what you're asking (possibly due to your transcription error). But I think you were asking why you can't just say $\mathbb{P}\left(\min \left\{U_1,\ldots,U_n\right\} \leq \frac{y}{n}\right) = \left(\mathbb{P}\left(U \leq...

Re: Statistics The integral you were thinking about is actually the "complementary CDF" or "survival function" (though the limits aren't right and I realised I misinterpreted your query before seeing your addendum), which is the probability Y_n > y. It's actually neater here find this survival...

Re: University Statistics Discussion Marathon Do you mean the value of the sum? (The "p" He-Mann used was referring to the success probability parameter, which is 0.5.)

Re: Statistics $\noindent If you want a ``by inspection'' method, you can observe the new region will be $y_{2} > 0$ and $y_{1} > -\frac{1}{2}y_{2}$. This is essentially because $Y_{2} = X_{2}$, so the range of values for $y_{2}$ is just the same as $x_{2}$, so is $y_{2} > 0$. For $Y_{1}$, we...

Re: Statistics $\noindent We just need to find what the region for the original sample space, which is $\Omega := \left\{ (x_{1}, x_{2}) \in \mathbb{R}^{2} : x_{1} > 0, x_{2} > 0\right\}$ (i.e. first quadrant), gets mapped to under the transformation in question, which is...

Re: MATH1131 help thread $\noindent You can say that since $p_{3}'(x) = p_{2}(x) > 0$ for all $x\in \mathbb{R}$, the function $p_{3}$ is increasing, so cannot have two real roots. (Alternatively, if it had two real roots, Rolle's Theorem would imply the existence of a real root of $p_{2}$, but...

Re: MATH1131 help thread $\noindent So for that last one, what you could do is decide what the function $f(t)$ is, what $a$ and $b$ should be for our purposes, calculate the derivative function $f'(t)$, and take $M$ to be the maximum value of $|f'(t)|$ on $[a,b]$. This would apply to the two...

Re: MATH1131 help thread $\noindent All of these Q's would use inequalities using Mean Value Theorem. Recall that for a function $f$ that is continuous on the closed interval $[a,b]$ and diffentiable on $(a,b)$, we have $f(b) = f(a) + f'(c) (b-a)$ for some $c \in (a,b)$. So for the last Q...

Re: MATH1131 help thread $\noindent Let $w \equiv z - 6 + i \iff z \equiv w + 6 - i$. Find all solutions $w$ to the equation $w^{3} = -27$. Then the solutions for $z$ in terms of the solutions for $w$ are given by $z = w + 6 - i$.$

Re: Statistics You integrate or sum the joint density function over all values of the other variable(s).

$\noindent The approach should be essentially as follows. Consider slicing the solid into thin slices parallel to the base. We know that a small slice between height $h$ and $h+\delta h$ has volume $\delta V \approx x^{2}(h)\delta h$ (where $x^{2}(h)\equiv (x(h))^{2}$), so the mass of solid in...

Re: University Statistics Discussion Marathon $\noindent I haven't checked your numerical calculations but yes that's the right approach, i.e. standardise the random variable using $Z = \frac{X -\mu}{\sigma}$ so that $Z$ becomes standard normal, correspondingly change the inequality bounds...

Re: Statistics $\noindent Depends on the question at hand. There might be faster methods sometimes.$ $\noindent For example, another possible approach is to compute the CDF of $Z:= X+Y$, $F_{Z}(z) := \mathbb{P}\left(X+Y \leq z\right)$, and then differentiate this wrt $z$ to get the PDF. (Or...