Random Inequalities

In this problem we look at two general 'random inequalities'. You can use the distribution maker interactivity to create distributions to try to solve the two parts of the problem.

Part 1
Markov's inequality tells us that the probability that the modulus of a random variable X exceeds any random positive number a is given by a universal inequality as follows:

$$ P(|X|\geq a) \leq \frac{E(|X|)}{a^{??}} $$

Part 2
Another important general statistical result is Chebyshev's inequality , which says that

$$ P(|X-\mu|\geq k\sigma)\leq \frac{1}{k^2} $$ where $\mu$ and $\sigma$ are the mean and standard devitation of the distribution $X$ respectively. This is true for any distribution and any positive number $k$. Can you make a probability distribution for which the inequality is exactly met when $k=2$? In other words, use the distribution maker to create a distribution $X$ for which $$ P(|X-\mu|\geq 2\sigma)=\frac{1}{4} $$