Home About us Mathematical Epidemiology Rweb EPITools Statistics Notes Web Design Contact us Links |

> Home > Statistics Notes > Probability > Expectation Values (3)

We will need to denote the expectation value of X by EX and the expectation value of Y by EY.

Now it is time to consider a third random variable. Now we need to take X, subtract off the expectation value of X itself from the values of X, and square that. What does this tell us? Remember that the expectation value is the "center of gravity" of the distribution; the average of large samples taken from X will be close to this expectation value of X. So when I subtract the expectation of X from the values of X, and square it, we have the squared deviation of X from the expectation value of X itself. So we define

We will keep using the binomial example (it makes the calculations simpler). We have been using the value N=3 and p=0.6; we found that the expectation value is 1.8=

How can we calculate the expectation value of U, which we will denote EU? We take all the values of U, multiply each by its probability, and add it all together. We will have

Try this computer experiment. Return to the binomial example I worked out. The expected value of the squared deviation from the mean is 0.72. If I calculate a lot of binomial random variables, subtract off the expected value of the random variable itself, square what I get, and average all those answers, I ought to get something close to 0.72, because the average of a lot of things ought to be close to the expectation value for those things. The average of a lot of values of a random variable should be near the expectation value of the random variable; the average of a lot of squared values for the random variable should be near the expected value of the squared random variable, and the average of a lot of squared deviations from the expectation value of a random variable should be near the expected squared deviation from the expectation value of the random variable.

```
> u1 <- rbinom(10000,size=3,prob=0.6)
```

> u2 <- u1-3*0.6

> u3 <- u2*u2

> mean(u3)

The expected value of the squared deviation from the expectation value of the random variable itself is called the

On to variance.

Return to statistics page.

Return to probability page.

Return to stochastic seminar.

All content © 2000 Mathepi.Com (except R and Rweb).

About us.