1. Chebychev, Hoffding, Chernoff

2. Histo 1
X = 2*Bin(300,1/2) – 300
E[X] = 0

3. Histo 2
Y = 2*Bin(30,1/2) – 30
E[Y] = 0

4. Histo 3
Z = 4*Bin(10,1/4) – 10
E[Z] = 0

5. Histo 4
W = 0
E[W] = 0

6. A natural question:
Is there a good parameter that allow to distinguish between these distributions?
Is there a way to measure the spread?

7. Variance and Standard Deviation
The variance of X, denoted by Var(X) is the mean squared deviation of X from its expected value m = E(X):
Var(X) = E[(X-m)2].
The standard deviation of X, denoted by SD(X) is the square root of the variance of X.

8. Computational Formula for Variance
E[ (X-m)2] = E[X2 – 2m X + m2]
E[ (X-m)2] = E[X2] – 2m E[X] + m2
E[ (X-m)2] = E[X2] – 2m2+ m2
E[ (X-m)2] = E[X2] – E[X]2

9. Properties of Variance and SD

10. Markov

12. Chebyshev’s Inequality
Theorem: X is random variable on sample space S, and P(X=r) it’s
probability distribution. Then for any positive real number r:
(proof in book)
In words: the probability of finding a value of X farther away from the mean
than r is smaller than the variance divided by r^2. r

13. Example:

14. Proof

15. Popular form (same)
“k standard deviations”

16. Example

18. Hoffding (1963)
Let be random variables that for

20. ,
Chernoff

21. Example