# Tutor profile: Adrian K.

## Questions

### Subject: Pre-Calculus

Consider the quadratic equation $f(x) = ax^2 + bx - c$ where $a, b, c$ are all positive constants. Show there exists at least one real root.

To show there exists at least one real root, we simply need to show the graph is zero for some value of $x$. First note that by construction, $f(0) = a\times 0^2 + b\times 0 - c= - c < 0$. So that the function is strictly negative at $x = 0$. Similarly, note that since $a$ and $b$ are positive, the function goes to infinity as $x$ goes to infinity, $\lim_{x\to \infty} f(x) = \lim_{x\to \infty} ax^2 + bx - c = \infty$ So we have shown that $f(x) = 0$ at $x=0$ and goes to positive infinity as $x \to \infty$. Since $f(x)$ is clearly continuous, this implies it must cross the $x$ axis at some point, and hence a real root exists.

### Subject: Pre-Algebra

An apple costs $x$ dollars. A watermelon is twice as expensive as an apple. A banana is $\frac{2}{3}$ as expensive as a watermelon. How expensive is a banana compared to an apple?

Let $y$ be the cost of a pineapple, and $z$ the cost of a banana. We wish to find the number $A$ such that $z = A x$. We are told a watermelon is twice as expensive as an apple. This can be written mathematically as $y = 2x$. Similarly, we are told a banana is two-thirds as expensive as a pineapple, and this can be written mathematically as $z = \frac{2}{3} y$ To find $A$, we simply substitute the first equation for $y$ (in terms of $x$) in the second equation. This gives $z = \frac{2}{3} y = \frac{2}{3} \times 2x = \frac{4}{3} x$. Thus, a banana is four-thirds as expensive as an apple.

### Subject: Statistics

Let $X_1, X_2, \dots, X_n$ be a random sample from an unknown distribution with finite mean $\mu$ and variance $\sigma^2$. Find the asymptotic distribution of $\sqrt{n}s_n^2$ where $s_n^2=\frac{1}{n}\sum_{i=1}^n (X_i - \bar{X}_n)^2$.

We write $\sum_{i=1}^n( X_i-\bar{X}_n)^2 = \sum_{i=1}^n (X_i-\mu)^2 + n(\bar{X}_n-\mu)^2$ The left hand term is the summation of identically distributed random variables. Thus we have $\sqrt{n}s_n^2 = \frac{1}{\sqrt{n}}\sum_{i=1}^n (X_i-\mu)^2 + \sqrt{n}(\bar{X}_n-\mu)^2$ The left-hand term is asymptotically normal by the central limit theorem. The right hand term converges to zero in probability since we can write $\sqrt{n}(\bar{X}_n-\mu)^2 = \frac{1}{\sqrt{n}} (\sqrt{n}(\bar{X}_n-\mu))^2$ and $\sqrt{n}(\bar{X}_n-\mu)$ is asymptotically normal by the central limit theorem. It then follows by Slutsky's theorem that $\sqrt{n}s_n^2$ is asymptotically normal.

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