Unit 10: Inequalities and Limit Theorems
Bounds that hold for any distribution at all, and the two theorems that make averages and sums predictable no matter the underlying distribution.
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10.1 Markov's Inequality
- How it worksHow does Markov's inequality turn a nonnegative random variable's mean into a bound on a tail probability?
- Worked exampleFind the Markov bound on P(X is at least 20) for a nonnegative random variable with mean 5.
- Common mistakeWhy doesn't Markov's inequality need to know anything about the shape of the distribution to work?
Your turn1 of 2
Find the Markov bound on P(X is at least 38) for a nonnegative random variable with mean 7.10.2 Chebyshev's Inequality
- How it worksHow does applying Markov's inequality to the squared deviation from the mean produce Chebyshev's bound?
- Worked exampleFind the Chebyshev bound on P(X is at least 3 away from its mean) for a random variable with variance 4.
- Going furtherWhat happens to a Chebyshev bound as the number of standard deviations away from the mean grows?
Your turn1 of 2
Find the Chebyshev bound on P(X is at least 7 away from its mean) for a random variable with variance 14.10.3 The Cauchy-Schwarz and Jensen Inequalities
- How it worksHow does Cauchy-Schwarz bound E[XY]2 by the product of E[X2] and E[Y2]?
- Worked exampleFind whether Jensen's inequality shows E[1/X] is at least or at most 1/E[X] for a positive random variable.
- Common mistakeWhy doesn't E[g(X)] equal g(E[X]) whenever g is a convex function like x2?
Your turn1 of 2
Two random variables satisfy E[X2] = 14 and E[Y2] = 13. Find the Cauchy-Schwarz upper bound on E[XY].10.4 The Law of Large Numbers
- How it worksHow does Chebyshev's inequality applied to the sample average prove the law of large numbers?
- Worked exampleFind how close a sample average of 10000 fair coin flips is likely to land to 0.5, using its standard deviation.
- Common mistakeWhy doesn't the law of large numbers mean a long run of heads makes tails 'due' to even things out?
- Common mistakeWhy doesn't the law of large numbers claim the absolute difference between heads and tails shrinks to zero?
- Going furtherWhat happens to the sample average's spread around the true mean as n grows from 100 to 1000000?
Your turn1 of 2
Using Chebyshev's inequality, find an upper bound on the probability that the sample average of 35124 fair coin flips lands more than 0.0684 away from 0.5.10.5 The Central Limit Theorem
- How it worksHow does standardizing a sum of many independent random variables make its distribution converge to standard Normal?
- Worked exampleFind the approximate probability that the average of 100 independent rolls of a fair die exceeds 3.6.
- Worked exampleFind the approximate probability that the sum of 50 independent Exponential(1) random variables exceeds 55.
- Common mistakeWhy doesn't the central limit theorem require the underlying population itself to be Normally distributed?
- Going furtherWhat happens to the accuracy of the CLT's Normal approximation as the sample size grows from 5 to 500?
Your turn1 of 2
Find the z-score used in the Normal approximation for the average of 363 independent rolls of a fair die exceeding 3.989 (die mean 3.5, variance 35/12).10.6 The Chi-Square Distribution
- How it worksHow does summing k independent squared standard Normals build the Chi-Square distribution with k degrees of freedom?
- Worked exampleFind the mean and variance of a Chi-Square distribution with 5 degrees of freedom.
- Going furtherWhat happens to the Chi-Square distribution's shape as its degrees of freedom grows large?
Your turn1 of 2
Find the variance of a Chi-Square distribution with 35 degrees of freedom.10.7 The Student-t Distribution
- How it worksHow does dividing a standard Normal by the square root of a scaled Chi-Square build the Student-t distribution?
- Worked exampleFind how the Student-t distribution with 3 degrees of freedom compares with the standard Normal at z equal to 2.
- Common mistakeWhy doesn't the Student-t distribution collapse to exactly the standard Normal at every finite degrees of freedom?
Your turn1 of 2
Find the variance of a Student-t distribution with 22 degrees of freedom (df greater than 2).10.8 The F Distribution
- How it worksHow does dividing two independent scaled Chi-Square random variables build the F distribution?
- Worked exampleFind the degrees of freedom of an F distribution formed from a Chi-Square with 4 degrees of freedom over one with 10.
- Going furtherWhat happens to the F distribution as both of its degrees-of-freedom parameters grow very large?
Your turn1 of 2
An F distribution is formed from a Chi-Square with 15 degrees of freedom over one with 21 degrees of freedom (each divided by its own degrees of freedom). Find the mean of this F distribution (valid when d2 is greater than 2).