Expected values
- Expected values are useful cor characterizing a distribution
- The mean is a characterization of its center
- The variance and standard deviation are characterizations of how spread out it is
- Our sample expected values (the sample mean and variance) will estimate the population versions
The sample mean
- The sample mean estimates this population mean
- The center of mass of the data is the empirical mean \[
\bar X = \sum_{i=1}^n x_i p(x_i)
\] where \(p(x_i) = 1/n\)
Example
Find the center of mass of the bars
The center of mass is the empirical mean
What about a biased coin?
- Suppose that a random variable, \(X\), is so that \(P(X=1) = p\) and \(P(X=0) = (1 - p)\)
- (This is a biased coin when \(p\neq 0.5\))
- What is its expected value? \[
E[X] = 0 * (1 - p) + 1 * p = p
\]
Example
- Suppose that a die is rolled and \(X\) is the number face up
- What is the expected value of \(X\)? \[
E[X] = 1 \times \frac{1}{6} + 2 \times \frac{1}{6} +
3 \times \frac{1}{6} + 4 \times \frac{1}{6} +
5 \times \frac{1}{6} + 6 \times \frac{1}{6} = 3.5
\]
- Again, the geometric argument makes this answer obvious without calculation.
Continuous random variables
- For a continuous random variable, \(X\), with density, \(f\), the expected value is again exactly the center of mass of the density
Example
- Consider a density where \(f(x) = 1\) for \(x\) between zero and one
- (Is this a valid density?)
- Suppose that \(X\) follows this density; what is its expected value?
Facts about expected values
- Recall that expected values are properties of distributions
- Note the average of random variables is itself a random variable and its associated distribution has an expected value
- The center of this distribution is the same as that of the original distribution
- Therefore, the expected value of the sample mean is the population mean that it’s trying to estimate
- When the expected value of an estimator is what its trying to estimate, we say that the estimator is unbiased
- Let’s try a simulation experiment
Averages of x die rolls
Sumarizing what we know
- Expected values are properties of distributions
- The population mean is the center of mass of population
- The sample mean is the center of mass of the observed data
- The sample mean is an estimate of the population mean
- The sample mean is unbiased
- The population mean of its distribution is the mean that it’s trying to estimate
- The more data that goes into the sample mean, the more concentrated its density / mass function is around the population mean