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LECT RE 18: Inequalities ,, co . vergence, nd the . Weak Law o arge N m e s • Inequalities - bound P(X > a) based on lim·ted information about a d·str·bution Markov inequality (based on the mean) Chebyshev . inequality (based on . the mean and var·ance) • WL . X,X1, ... ,Xn . i.i.d. X . 1 +··· Xn ----- …

lecture 14: markov and chebyshev’s inequalities 3 Let us apply Markov and Chebyshev’s inequality to some common distributions. Example: Bernoulli Distribution The Bernoulli distribution is the distribution of a coin toss that has a probability p of giving heads. Let X denote the number of heads. Then we have E[X] = p, Var[X] = p p2.

Thus, we conclude P(X≥a)≤EXa,for any a>0. We can prove the above inequality for discrete or mixed random variables similarly (using the generalized PDF), so ...

lecture 14: markov and chebyshev’s inequalities 3 Let us apply Markov and Chebyshev’s inequality to some common distributions. Example: Bernoulli Distribution The Bernoulli distribution is the distribution of a coin toss that has a probability p of giving heads. Let X denote the number of heads. Then we have E[X] = p, Var[X] = p p2.

Markov’s Inequality, Pr(Y a2) E[Y] a2 = E ( X[ ])2 a2 = Var[X] a2: Example. Again consider the fair coin example. Recall that Xdenotes the number of heads, when nfair coins are tossed independently. We saw that Pr(X 3n 4) 2 3, using Markov’s Inequality. Let us see how Chebyshev’s Inequality can be used to give a much stronger bound on ...

One use of Markov's inequality is to use the expectation to control the probability distribution of a random variable. For example, let X be a non- negative ...

Markov’s Inequality, Pr(Y a2) E[Y] a2 = E ( X[ ])2 a2 = Var[X] a2: Example. Again consider the fair coin example. Recall that Xdenotes the number of heads, when nfair coins are tossed independently. We saw that Pr(X 3n 4) 2 3, using Markov’s Inequality. Let us see how Chebyshev’s Inequality can be used to give a much stronger bound on ...

P(X ≥ a) ≤. E(X) a provided E(X) exists. For example, Markov's inequality tells us that as long as X doesn't take negative values, the probability that X is ...

Today we are going to talk more about the probability that a random variable deviates from its expectation. We have already seen examples ...

To obtain an even tighter bound than Markov's and Chebyshev's, we need an additional level of independence. For example, if X1 · X2 = X3 (all ...

Probability Bites Lesson 39 Markov and Chebyshev Inequalities Rich Radke Department of Electrical ...

take large values, and will usually give much better bounds than Markov’s inequality. Let’s revisit Example 3 in which we toss a weighted coin with probability of landing heads 20%. Doing this 20 times, Markov’s inequality gives a bound of 1 4 on the probability that at least 16 ips result in heads. Using Chebyshev’s inequality, P(X 16 ...

Markov and Chebyshev Inequalities , Law of Large Numbers Parametric and non-parametric estimation Example of the Law of Large Numbers Empirical mean and covariance Central Limit Theorem, Change of Variables and Monte Carlo, Estimating p by MC, Accuracy of MC, Poisson Process, Approximating

Again, if we didn't know the PMF/PDF of what we cared about, we could use the sample mean as a good estimate for the true mean (by the Law of Large Numbers from ...

Markov and Chebyshev’s Inequalities; and Examples in probability: the birthday problem Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 7) 1 / 12

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Markov’s inequality is tight, because we could replace 10 with tand use Bernoulli(1, 1/t), at least with t 1. Proving the Chebyshev Inequality. 1. For any random variable Xand scalars t;a2R with t>0, convince yourself that Pr[ jX aj t] = Pr[ (X a)2 t2] 2.

Exercise 10. Like we did in Example 4 for Markov’s inequality, prove that Chebyshev’s inequality is tight: nd a probability distribution for X and a value asuch that P(jX E(X)j a) = Var(X) a2. (Hint: This random variable will take only three values.)

The Markov and Chebyshev Inequalities We intuitively feel it is rare for an observation to deviate greatly from the expected value. Markov’s inequality and Chebyshev’s inequality place this intuition on firm mathematical ground.

The Chebyshev inequality • Random variable X, with finite mean . I" and variance a . 2 • "If the variance is small, then X is unlikely to be too far from the mean" Chebyshev inequality: Markov inequality: If X > 0 and . a> 0, then P(X > a) < E[X] a . 4

Markov and Chebyshev's Inequalities; and. Examples in probability: the birthday problem. Kousha Etessami. U. of Edinburgh, UK.

6.2.2 Markov and Chebyshev Inequalities. = a P ( X ≥ a). P ( X ≥ a) ≤ E X a, for any a > 0. We can prove the above inequality for discrete or mixed random variables similarly (using the generalized PDF), so we have the following result, called Markov's inequality .

6.2.2 Markov and Chebyshev Inequalities. = a P ( X ≥ a). P ( X ≥ a) ≤ E X a, for any a > 0. We can prove the above inequality for discrete or mixed random variables similarly (using the generalized PDF), so we have the following result, called Markov's inequality .

The Markov and Chebyshev Inequalities. We intuitively feel it is rare for an observation to deviate greatly from the expected value. Markov’s inequality and Chebyshev’s inequality place this intuition on firm mathematical ground. I use the following graph to remember them. Here, \(n\) is some positive number.

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bounds, such as Chebyshev's Inequality. Theorem 1 (Markov's Inequality) Let X be a non-negative random variable. Then,. Pr(X ≥ a) ≤.

tk . For many random variables (we will see some examples today), the moment generating function will exist in a neighborhood around 0, i.e ...