Questions about Poisson distribution?

Question

1) I've been told that specifically for a poisson distribution the mean is equal to the variance however I have seen many instances where it is only approximately equal and the poisson model has been used so I'm a little confused.Also out of interest does this mean there are other distribution models where the variance and mean are equal/approximately equal

2) Just to clear things us on a calculator display (specifically cassio classwiz) does s^2x denote variance and does σ^2 also denote variance (and what does σ^2x on a calculator display mean??)

3) I thought you should never round E(X) (or x̄) even for discrete data 

however the answers provided to the following question where given as integers so if anyone knows why:

A typesetter makes 1500 mistakes in a book of 500 pages on how many pages would you expect to find i) 0 mistakes  e.t.c (answer to this first question just given as 25)

Answer 1

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1.  In a Poisson distribution, the mean equals the variance.  But any probability distribution is only an approximation of reality.  If you have data where the mean and variance are close, that is some evidence you should consider the use of a Poisson.  Better evidence would be to use a chi-square test comparing your actual data with what a Poisson expects.  If you decide Poisson is not a good fit to your data, consider something else;  negative binomial is one choice.

2.  I don't have a Casio, but s^2 sometimes denotes sample variance with an n-1 denominator, in contrast to population variance with an n denominator.

3.  The rounding rule is don't round intermediate calculations.  But you may round the final answer.  If the problem is to calculate how  many pages, the answer should be an integer.