Find an example of a random variable with finite expectation and unbounded variance. Give a clea

Find an example of a random variable with finite expectation...

Key Concepts

Moment Conditions

Moment conditions refer to the requirements for the finiteness of moments (like the mean and variance) of a distribution. In many distributions, especially those with heavy tails, lower-order moments (such as the first moment or mean) may be finite while higher-order moments (such as the second moment or variance) are infinite. Understanding these conditions is crucial when analyzing the properties of a given random variable.

Expectation

The expectation of a random variable is its average or mean value, computed as a weighted sum (or integral) of all possible outcomes using their probabilities. It provides a measure of central tendency and only requires that the first moment of the random variable exists (i.e., is finite) for it to be well-defined.

Variance

The variance measures the spread or dispersion of a random variable around its mean by averaging the squared deviations. It requires the second moment to exist. In some cases, even if the expectation is finite, the variance (second moment) can be unbounded or infinite, especially if the random variable exhibits heavy tails.

Heavy-Tailed Distribution

Heavy-tailed distributions are those in which the probability of extreme events decays slowly (often following a power law). These distributions can have finite lower-order moments (like the mean) while having infinite higher-order moments (like the variance) if the tail decays sufficiently slowly. Choosing a heavy-tailed distribution with a tail exponent between 2 and 3 is a typical way to construct an example with finite expectation but infinite variance.

Transcript

00:01 Here we want to show that the variance of two events is not necessarily equal to the sum of their independent variances.

00:27 Okay, so take, for example, a coin flip.

00:31 So the probability of getting heads is going to be equal to the probability of getting a tails, which is one half, right? and so we get that the probability of getting a heads plus the probability of getting a tails is going to be equal to one.

00:51 It's one half plus one half.

00:54 So we consider the expected values.

00:58 Okay, so the expected value of getting a heads would be equal to the sum of the product of each possibility with its probability, right? so it's going to be equal to the sum of each possibility multiplied with its probability.

01:41 Okay, so that's going to end up being either you didn't get it with a probability of one -half, or you got your heads with a probability of one -half is equal to one -half.

01:56 And by the exact same argument, you will see that this is the expected value of getting a, tails as well.

02:08 Okay, and the expected value of both events would then be equal to the, you know, we use the exact same definition, but with it, by adding them together.

02:34 So this is going to end up being 1 times 1 is equal to 1.

02:43 And if you want to look at the expected value of a squared or for x squared, so this is going to help us set up calculating our variances, right? because by definition, the variance, of x is equal to the expected value of x squared minus the expected value of x squared.

03:21 Okay, so we're going to calculate this, and this is basically the same.

03:29 It's up here, except we're going to square our x, right? so this is going to end up being equal to 0 squared times 1 half plus 1 squared times 1 1 half is equal to 1 half and again by the same argument and see if this is equal to the expected value of getting a tails squared now i want to consider the expected value of getting the heads and tails squared okay and by the same argument it's going to be the same as up here except with a where we are squaring that value as well and so then we get 1 squared times 1 1 is equal to 1...

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