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## 'Random Inequalities' printed from http://nrich.maths.org/

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