1. Chernoff bounds The Chernoff bound for a random variable X is
obtained as follows: for any t >0,
Pr[X  a] = Pr[etX  eta] ≤ E[etX ] / eta
Similarly, for any t <0,
Pr[X  a] = Pr[etX  eta] ≤ E[etX ] / eta
The value of t that minimizes E[etX ] / eta gives the best possible bounds.

2. Moment generating functions
Def: The moment generating function of a random variable X is MX(t) = E[etX].
E[Xn] = MX(n)(0) , which is the nth derivative of MX(t) evaluated at t = 0.
Fact: If MX(t)= MY(t) for all t in (-c, c) for some
c > 0, then X and Y have the same distribution.
If X and Y are independent r.v., then
MX+Y(t)= MX(t) MY(t).

3. Chernoff bounds for the sum of Poisson trials
Poisson trials: the distribution of a sum of independent 0-1 random variables, which may not be identical.
Bernoulli trials: same as above except that all the random variables are identical.
Xi:i=1…n, mutually independent 0-1 r.v. with
Pr[Xi=1]=pi.
Let X =X1+…+Xn and E[X] =μ=p1+...+pn.
MXi(t) =E[etXi] = piet +(1-pi) = 1 + pi (et -1)
≤ exp (pi (et -1) ).

4. Chernoff bound for a sum of Poisson trials
MX(t) = MX1(t) MX2(t) .... MXn(t)
≤ exp{(p pn)(et -1)} = exp{(et -1)μ} T h e o r m L t X = 1 + n , w ; : a i d p l s u c P [ ] f 2 . ( ) y
> 3 R 6

5. Proof: B y M a r k o v i n e q u l t , f > w h P [ X ¸ ( 1 + d ) ¹w h P [ X ( 1 + d ) ] = E . F s T p 2
< 3 g m b c R 6 5 H

6. Similarly, we have: T h e o r m L t X = P , w ; : a d p s l u c [ ] .n i 1 , w ; : a d p s l u c [ ] . E f
< ( ) 2 C y F j 3

7. Example: Let X be the number of heads of n independent fair coin flips
Example: Let X be the number of heads of n independent fair coin flips. Applying the above Corollary, we have: P r [ j X n = 2 p 6 l ] e x ( 1 3 ) : 4 B y C h b s v i q u a t , . E V w

8. Application: Estimating a parameter
Given a DNA sample, a lab test can determine if it carries the mutation. Since the test is expensive and we would like to obtain a relatively reliable estimate from a small number of samples. Let p be the unknown parameter that we are looking for estimation.
Assume we have n samples and X=p~ n of these samples have the mutation. For sufficient large number of samples, we expect p to be close to p~. D e f : A 1 r c o n d i t v a l p m s [ ~ ; + ] u h P 2 .

9. A m o n g t h e s a p l , w ¯ d X = ~ u i . W r f c P [ 2 ¡ ; + ] ( )1 b E I v :
<
> T 3 S

10. Better bounds for special casesh e o r m L t X = 1 + n , w ; : a i d p v b l s P [ ] 2 . F y
> f E ! ( ) u g S B

11. Better bounds for special casesy L e t X = 1 + n , w h ; : i d p m v b s P [ ] 2 . F
> j Y ( ) f g E T u 4

12. Better bounds for special casesy L e t Y = 1 + n , w h ; : i d p m v b s P [ ] 2 . E ( ) F
> A c S g G x - f u T q j 4

13. Proof of set balancing:h i - w A b a = ( ; 1 m ) n d s u p k . I 4 l , c y j v S
> z Z B C [ ] 2