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I think it's mainly the x and y parts that are confusing you guys? If you want, you can rewrite the partials of f as f1 and f2, because that's what they're referring to with those partials, it's just a bit of abuse of notation I guess what they've done.

$\noindent If $x$ and $y$ are fixed, then $f\left(tx,ty\right)$ is just a function of $t$, and so then we can write $\frac{\mathrm{d}}{\mathrm{d}t}$. Further, $f\left(tx,ty\right)=g\left(x,y,t\right)$, so the derivative of $f$ wrt $t$ is just that of $g$ wrt t (to help, you might want to write...

Second one is also an application of chain rule. $Let $ u(x,y) = f(\alpha x + \beta y)$. Then via chain rule,$ $\noindent $u_x= f^{\prime} (\alpha x + \beta y) \cdot \alpha$ and $u_y = f^\prime (\alpha x + \beta y) \cdot \beta$ (using subscript notation for partial derivatives). If we sub...

For the first one, partially differentiate the homogeneity equation wrt t (that's how you get n to pop out) and then sub. in t = 1 after that. To differentiate the LHS of that wrt t, $\noindent Let $u=xt$, $v=yt$ and $z = f(xt,yt) = f(u,v)$. Then $u_t = x$ and $v_t = y$ (subscripts denote...

By definition of limits, since f(x)/g(x) -> 0 as x -> oo, we know for ANY eps > 0, we have |f(x)/g(x) - 0| < eps <=> f(x)/g(x) < eps, for all x greater than some M (i.e. for all x sufficiently large). In particular, we can choose eps = 1 and get that f(x)/g(x) < 1 <=> f(x) < g(x) for all x...

QUESTION (iii) $\noindent Let $y>0$, the cdf of $Y$ be $G$ and the cdf of $X$ be $F$. Then $ $$\begin{align*}G(y) &= \mathbb{P}\left(Y \leq y\right) \\ &= \mathbb{P}\left(e^X \leq y \right) \\ &= \mathbb{P}\left(X \leq \ln y \right) \quad (\text{exponential function is increasing}) \\ &=...

QUESTION (ii) $\noindent For questions like this, the important thing is to define a useful random variable(s), write down or work out its distribution, and work out what the question is asking of us in terms of our random variable.$ $\noindent Let $X$ be the number of claims received by...

QUESTION (i) (I said 'test' a lot below, I'm referring to the method for detecting illegal trades by that.) • $\noindent $I$ be the event a trade is illegal (so $I^c$ is the event a trade is legal)$ • $\noindent $T$ be the event that the \textsl{test} says a randomly chosen trade is illegal...

$\noindent For the first one, it rearranges to $\tan \alpha = -\frac{1}{\sqrt{3}}$. Since $\alpha$ is obtuse, we want the solution in the range $90^\circ < \alpha < 180^\circ$. Since $\tan 30^\circ = \frac{1}{\sqrt{3}}$, we have $\tan 150^\circ = -\frac{1}{\sqrt{3}}$, so $\alpha = 150^\circ$, or...

$\noindent The last question here is basically just an integration exercise.$ $\noindent Let's assume the random variable is defined on the whole real line (so integrals we do will be from $-\infty$ to $+\infty$). The first step is to find the constant of proportionality $c$ such that...

Here's some hints $\noindent In the Binomial model, the probability of an arbitrarily chosen person from this large population being colour-blind is $p=0.01$. If we choose a sample (independently and so on) of size $n$ which is much smaller than the total population size, this is to a good...

No, although it may be in the future (I hear they're planning on bringing Statistics into Maths Extension 2 later).

I would probably answer your questions by saying "by inspection".

Yeah the chance of having a normal r.v. be 10 standard deviations away from its mean is basically 0, so it makes no real difference.

Here is where: .

Didn't I already answer most of them in two words ("by inspection")?

Yeah that's basically what needs to be done. Although, it might be better to find Pr((100/150) < X-bar ≤ 1) instead, since X-bar can't exceed 1, but if you use the Normal approximation, you make it as though it can exceed 1, so you need to account for that by making sure you compute Pr((100/150)...

It's just substituting numbers into a formula. Which part did you get stuck on?

Yep, and P(1) = 1 – 2 + 3 – 1 = 1, so the answer is indeed 8 as wu345 said.

If you know the formula, why do you need help with those Q's? Or do you know how to do them now?