How is Chernoff bound calculated?

How is Chernoff bound calculated?

Using Chernoff bounds, find an upper bound on P(X≥αn), where p<α<1. Evaluate the bound for p=12 and α=34. For X∼Binomial(n,p), we have MX(s)=(pes+q)n, where q=1−p.

What is an upper bound in probability?

In probability theory, Markov’s inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant.

How do you calculate lower bound probability?

Let X be the number of heads one would obtain in 140 flips of a fair coin. Use Chebychev’s Inequality to find a lower bound on the probability P(60kσ)≤1/k2 for k>0, where σ2 is the variance of X.

Is Chernoff better than chebyshev?

Chernoff gives a much stronger bound on the probability of deviation than Chebyshev. This is because Chebyshev only uses pairwise independence between the r.v.s whereas Chernoff uses full independence.

What is MGF of normal distribution?

This gives us a much better way to analytically tract the probability distribution of x (compared to the convolution approach). MGF for the Normal Distribution Here we assume that the random variables x follows a normal distribution.

What is tail bound?

In probabilistic analysis, we often need to bound the probability that a. random variable deviates far from its mean. There are various formulas. for this purpose. These are called tail bounds.

What is upper bound and lower bound in probability?

Upper and lower bounds are derived for the probability that at least r events occur and exactly r events occur. It is assumed that the probabilities of intersection of at most m events are known.

What is upper and lower bound probability?

Upper and lower probabilities are representations of imprecise probability. Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, this method uses two numbers: the upper probability of the event and the lower probability of the event.

What is a lower bound of probability?

The lower bound is the smallest value that would round up to the estimated value. The upper bound is the smallest value that would round up to the next estimated value. For example, a mass of 70 kg, rounded to the nearest 10 kg, has a lower bound of 65 kg, because 65 kg is the smallest mass that rounds to 70 kg.

What is the mgf of uniform distribution?

The moment-generating function is: For a random variable following this distribution, the expected value is then m1 = (a + b)/2 and the variance is m2 − m12 = (b − a)2/12.

What is the mgf of geometric distribution?

The something is just the mgf of the geometric distribution with parameter p. So the sum of n independent geometric random variables with the same p gives the negative binomial with parameters p and n. for all nonzero t. Another moment generating function that is used is E[eitX].