Chebyshev inequality If X is a random variable with mean u and variance o2,then 02 P(IX-川≥c)≤c2, for all c >0 6
Chebyshev inequality • If 𝑋 is a random variable with mean 𝜇 and variance 𝜎 2 , then 𝑃 𝑋 − 𝜇 ≥ 𝑐 ≤ 𝜎 2 𝑐 2 , for all 𝑐 > 0 6
Example 2:Uninformative case Let X be uniformly distributed in 0,4. Let us use the Chebyshev inequality to bound the probability that |X-2≥1. ·We haveσ2=16/12=4/3，and P(IX-2≥1)≤4/3 Which is uninformative. 7
Example 2: Uninformative case • Let 𝑋 be uniformly distributed in 0,4 . • Let us use the Chebyshev inequality to bound the probability that |𝑋 − 2| ≥ 1. • We have 𝜎 2 = 16/12 = 4/3, and 𝑃 |𝑋 − 2 ≥ 1 ≤ 4/3 Which is uninformative. 7