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Symmetry of Normal Distribution?

S

Snowy

Member
Hi,
Page 83 of the Notes:
Why do we have P(z-1.96<Z<1.96) and not just P(-1.96<Z<1.96)?
The solution to this question relies on the assumption of symmetry because S is normally distributed and Z is the standard Normal distribution, right?
Can we always assume symmetry when the random variable is normally distributed or only when we convert to the Standard Normal distribution eg what if S was Poisson or Geometric?
 
Hi Snowy

This first expression is a mistake. As you say, it should just be: P(-1.96<Z<1.96) = 0.95. Sorry to have caused confusion - I'll get it corrected.

If the distribution is normal, then it is symmetrical about the mean. If the distribution is Poisson or Binomial, then, under certain circumstances, we can approximate it by the normal distribution. The Central Limit Theorem says that, regardless of the distribution of X, the sample mean (X bar) is distributed approximately normally.
 
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