Answer by Calvin Khor for What is the intuition behind Chebyshev's Inequality...
Let's assume the usual setup of non-negative, measurable functions on a space with finite Lebesgue measure. Bounded functions are integrable, but there are many integrable functions that are not...
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Answer by Nick Alger for What is the intuition behind Chebyshev's Inequality...
It can be helpful to draw a picture:Here:the blue curve is $f(x)$,the base of the red box is the set $\{x \in E: f(x) \ge \lambda\}$,the height of the red box is $\lambda$.Chebyshev's inequality says...
View ArticleAnswer by Yakk for What is the intuition behind Chebyshev's Inequality in...
Flip it around, and cutting E for simplicity, we get:$$m\{ f(x) \ge \lambda \} * \lambda \leq \int f$$The measure over the region where $f$ is at least $\lambda$, times $\lambda$, isn't bigger than the...
View ArticleAnswer by littleO for What is the intuition behind Chebyshev's Inequality in...
You could think of it like this. At a birthday party, everyone eats a certain amount of cupcakes. The total number of cupcakes eaten is greater than or equal to $\lambda$ times the number of people who...
View ArticleAnswer by ir7 for What is the intuition behind Chebyshev's Inequality in...
In a probabilistic setting, this leads to an upper bound on the total of two tails of a distribution when the tails start at equal distances on either side of the mean:$$P(|X-\mu| \geq \lambda) =...
View ArticleAnswer by Ramiro for What is the intuition behind Chebyshev's Inequality in...
The essential point is that$$0 \leq \lambda \cdot 1_{\{x \in E \;|\; f(x) \geq \lambda \} } \leq f$$where $1_{\{x \in E \;|\; f(x) \geq \lambda \} }$ is the characteristic (indicator) function of $\{x...
View ArticleWhat is the intuition behind Chebyshev's Inequality in Measure Theory
Chebyshev's Inequality Let $f$ be a nonnegative measurable function on $E .$ Then for any $\lambda>0$,$$m\{x \in E \mid f(x) \geq \lambda\} \leq \frac{1}{\lambda} \cdot \int_{E} f.$$What exactly is...
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