Random Inequalities

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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:

$$ P(|X|\geq a) \leq \frac{E(|X|)}{a^{??}} $$
In this expression the exponent of the denominator on the right hand side is missing, although Markov showed that it is the same whole number for every possible distribution . Given this fact, experiment with the various distributions to find the missing value (??).

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 $$ P(|X-\mu|\geq 2\sigma)=\frac{1}{4} $$