Question

A question about the Bell Curve

Answer 1

\[e ^{-0.5(x-\mu)^2/\sigma^2}/\sigma \sqrt{2 \pi}\]
\[e ^{0.5(-(x-\mu)^2/\sigma^2)}/\sigma \sqrt{2 \pi}\]
\[e ^{i(x-\mu)/\sigma}/\sigma \sqrt{2 \pi}\]
\[(cos(x-\mu/\sigma)+isin(x-\mu/\sigma))/\sigma \sqrt{2 \pi}\]

Answer 2

The last equation graphs nothing like the first, and I can't see why, as they should be the same thing.

Answer 3

how did u get the third equation from the second one ? why there is iota there ?

Answer 4

^0.5 is the square root

Answer 5

\[\sqrt{-1}\sqrt{(x-\mu)^2}/\sqrt{\sigma^2}\]

Answer 6

WHat's the question here?

Answer 7

The first equation is the bell curve, but the second seems periodic, it will yield a completely different graph. How is this?

Answer 8

I guess you are confused by the 0.5 in your starting formula. If you write it in the form: \[1/(\sigma \sqrt{2\pi} ) \times e ^{-[(x-\mu)/(\sigma \sqrt{2} )]^2} \] and use \[i ^{2} = -1 \] like you suggested, you get:
\[1/(\sigma \sqrt{2\pi} ) \times e ^{[i^{2}(x-\mu)/(\sigma \sqrt{2} )]^2} \]
This is just another way of writing the bell formula. I fact it is a non-periodic real function.

Answer 9

You start with
\[ e ^{0.5(-(x-\mu)^2/\sigma^2)}/\sigma \sqrt{2 \pi} \]
and then re-write the 0.5 in the exponent as a square root.

But notice that the square root is around \( e^{stuff} \) and not around \(stuff\)
that is, the next step is
\[ \sqrt{e ^{(-(x-\mu)^2/\sigma^2)}}/\sigma \sqrt{2 \pi} \]
which is different than what you posted.