Looking at question 10.5 from the combined material pack, calculating σ^2 using E[X^2] seems to yield a different result to calculating it directly from s^2, even though σ^2 = s^2. Can some explain why this may be ?
Indeed you are confused. s² is the SAMPLE variance and is based on our sample of observations. σ² is the POPULATION variance is based on our probability model. We observe sample data that we believe to come from this model (though in reality we do not actually know the model underlining real life data - we just use guesses and go with the models that appear to best fit the data - but no model is a perfect fit to real life data). The method of moments estimate then makes the reasonable assumption that if the data comes from this model then the sample moments should be equal to the model moments. Now the reason why Σx²/n gives a different result to s² is that the s² formula divides by n-1. They would give the same estimates if we divided by n instead.
The solution seems to suggest dividing by either n or n-1 is acceptable so that was my main source of confusion. Is there one that's more appropriate?
Yeah, in CT3 they're very big on n-1 only. Whereas CT6 they tend to be much more dividing by n. So use that as your default.
