The Markov Inequality Let g(Y) be a function of the...

The Markov Inequality Let $g(Y)$ be a function of the continuous random variable $Y$, with $E(l g(Y) |)<\infty .$ Show that, for every positive constant $k.$ $$ P(|g(Y)| \leq k) \geq 1-\frac{E(|g(Y)|)}{k} $$ [Note: This inequality also holds for discrete random variables, with an obvious adaptation in the proof.]

The Markov Inequality Let $g(Y)$ be a function of the continuous random variable $Y$, with $E(l g(Y) |)<\infty .$ Show that, for every positive constant $k.$
$$
P(|g(Y)| \leq k) \geq 1-\frac{E(|g(Y)|)}{k}
$$
[Note: This inequality also holds for discrete random variables, with an obvious adaptation in the proof.]

Key Concepts

Random Variables

Random variables are functions that assign numerical values to the outcomes of a random process. They can be either continuous or discrete, and properties of random variables, such as their expectation and distribution, are central to probability theory. Comparisons and inequalities involving random variables help in understanding their behavior and in approximating or bounding probabilities.

Functions of Random Variables

Functions of random variables involve applying a mathematical function to a random variable, producing a new random variable with a transformed distribution. In the given context, the absolute value function applied to g(Y) ensures the resulting variable is non-negative, which is a critical condition for applying the Markov Inequality. Such transformations are common in probability theory to simplify analysis or to meet the requirements of specific theorems.

Markov Inequality

The Markov Inequality provides an upper bound on the probability that a non-negative random variable exceeds a specific value. In essence, it states that the probability that such a random variable is at least as large as a given positive constant is no more than the expected value of the random variable divided by that constant. This inequality is a fundamental tool in probability theory, used to make probabilistic statements when only the mean of the distribution is known.

Expectation

The expectation, or expected value, of a random variable is a measure of its central tendency, representative of the 'average' outcome. In the context of the Markov Inequality, the finiteness of the expectation is essential, as the inequality bounds the probability using this expected value. It reflects the integral (or sum, in the discrete case) of the variable’s values weighted by their probabilities.

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