MATH230 Week 02 - Assessed problems (coursework)

A random variable, $X$, has an expectation of $\mu$ and a variance of ${\sigma }^2$.

1. (a)

   Show that its skewness can be written as

   ${\displaystyle \frac{1}{{\sigma }^3}}\left(𝖤\left[X^3\right]-{\mu }^3\right)-3{\displaystyle \frac{\mu }{\sigma }}.$

2. (b)

   Hence derive an expression for $𝖤\left[X^3\right]$ in terms of $\mu$ and $\sigma$ for a random variable whose pdf or pmf is symmetric about $\mu$.

[marks: 5]

The lifetime, $X$, in days of a species of butterfly has a density of

(You discovered the value of $b$ in CW01.)

1. (a)

   Find the expectation of $X^a$ for $a>0$. What constraints are there on $a$ for the expectation to be finite?

2. (b)

   Write down the expectation and variance of $X$.

3. (c)

   Using the formula in Skew2, find the skewness of $X$.

[marks: 6]

Una and Ed have just phoned for a taxi. Una suggests modelling their waiting time as $\mathrm{𝖴𝗇𝗂𝖿}(a,b)$ but Ed believes $c+\mathrm{𝖤𝗑𝗉}(\beta )$ is better. Discuss the pros and cons of these options. Note: this question is not asking you to discuss how you would choose $a,b$ or $\beta$.

[marks: 4]

For a sufficiently smooth function, $f(x)$, Taylor expansion about some point, $\mu$, gives

for some $\eta (\mu ,x)$ between $\mu$ and $x$. Consider any function $f$ with $f^{\prime \prime }(x)\ge 0$ for all $x\in ℝ$ and show that $𝖤\left[f(X)\right]\ge f(𝖤\left[X\right])$. Hence relate $𝖤\left[X^2\right]$ to $𝖤\left[X\right]$ and $𝖤\left[e^X\right]$ to $𝖤\left[X\right]$.

[marks: 5]