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I think these were both answered on this page: http://community.boredofstudies.org/238/extracurricular-topics/350320/introductory-probability.html#post7154145 .

$\noindent Let $A$ be the set of all students who drink Brand A, and similarly define sets $B$ and $C$ (by `drink Brand A', we mean `at least' Brand A, to use the question's phrasing.). Let $U$ be the universal set. We are asked to find $|A^c \cap B^c \cap C^c|$. We are given: $|U| = 150, |A| =...

Another interesting read about moderation: .

No, that's not how it works. But the person who came first internally would get the highest external mark as their internal.

$\noindent We can't have $k=0$. Note $\frac{p_{k}}{p_{k-1}}\geq 1 \Longleftrightarrow \frac{\lambda}{k}\geq 1 \Longleftrightarrow k \leq \lambda$. So the max. $p_k$ occurs at that $k$ for which $k$ is the largest integer that is less than or equal to $\lambda$; in other words, at...

$\noindent Here's how to do your Question 3 (below).$ ''3. A particle is moving in a straight line. It starts from a fixed point O on the line and at time t second its velocity v m/s is given by v=2cost a. Find the first two times when the particle is at rest b. find the distance travelled...

Why would you be dead if you hadn't asked? You could've always asked your teacher or someone tomorrow if no-one had been able to help here, couldn't you?

Looks like no-one did yet. But that's a 4U type question (I think). Are you already up to 4U?

Ah. I'd thought eyeseeyou was referring to crossing the horizontal axis. Yeah, should only draw it for t ≥ 0.

$\noindent Did the answer sheet give the equation for $x(t)$? The one davidgoes4wce has is correct, and the graph indeed crosses the horizontal axis.$

Why not?

Practise heaps of easy (but easy-to-make-silly-mistakes-on) questions.

Yes, it's possible to get 95+ ATAR with those subjects.

$\noindent Here's how to do part (b).$ $Starting from $\mathbb{E} \left[X^2 \right]= \sum_{k=1}^{\infty} \frac{k\lambda^k \mathrm{e}^{-\lambda}}{\left(k-1\right)!}$, we use the change of variable $j=k-1\Longleftrightarrow k = j+1$. When $k=1$, $j=0$, and as $k\to \infty$, $j\to \infty$, so we...

To do part b), the change of variable should be l = k-1, rather than +1 (it's (the change of variables and limits of summation) done basically like how we'd do a change of variables in integration).

$\noindent Part (c) is the easy part if the other two are done, assuming you know or have shown that $\mathbb{E}\left[X\right] = \lambda$. In that case, $\mathrm{Var}\left[X\right] =\mathbb{E}\left[X^2\right]-\big{(}\mathbb{E}\left[X\right] \right\big{)}^2 = \lambda^2 + \lambda - \lambda^2 =...

$\noindent For the mode, $\frac{p_k}{p_{k-1}}\leq 1\Longleftrightarrow k\leq (n+1)p$ as you showed. So the $k$ with highest $p_k$ is the largest integer $k$ that is no larger than $(n+1)p$, which is precisely $\lfloor \left(n+1\right)p\rfloor$ (if $(n+1)p$ is non-integral, which is usually the...

I think a Moderator moved it a while back. It is now here: http://community.boredofstudies.org/1003/maths/350202/university-statistics-discussion-marathon.html .

I think it is meant to be interpreted as: Stage 1 (time 0 to 10): Tap's rate increases uniformly from 0 to 6/5 L/s. Stage 2 (time 10 to 10+T): Tap's rate stays flat at 6/5 L/s. (This is the stage where it is fully open.) Stage 3 (time 10+T to 20+T (i.e. (10+T) + 10)): Tap's flow rate...

We need to include a part in the R vs. t graph that slopes down to 0 (symmetrical to the initial increase). This is because we are told the rate goes down at a constant rate for the last 10 seconds. So using symmetry, the area under the decreasing part at the end will also be 6. Then the area...