Markov's Inequality - Intuitively and visually explained скачать в хорошем качестве

Markov's Inequality - Intuitively and visually explained

Intuitive and visual explanation of Markov’s inequality. Markov's inequality provides an upper bound to the probability that a non-negative random variable is greater than or equal to some positive constant (threshold) k. The inequality relates the ratio of the expected value of and the threshold to the probability of the random variable exceeding the threshold. Markov’s inequality makes no assumption of the type of distribution a random variable follows. It gives an upper bound for probability of observing a value above some constant k. This upper bound must be true for all distributions. Hence, the upper bound is conservatively high. The upper bound decreases as the threshold k increases. Intuitively, a random variable with a relatively small mean (compared to the threshold k), has lower probability of being high (i.e., exceeding the threshold). The strict equality in Markov’s inequality, is satisfied when there is a probability mass at the threshold value, with the rest of the probability mass being at 0.