Properties of Normal Distribution

Question

Describe if the following statements are true or false. Explain why.

1. For any normal distribution, the mean, median, and mode are equal.
2. The line of symmetry for a normal distribution is x=μ.
3. The y-axis is a vertical asymptote for all normal distributions.
4. The inflection points for any normal distribution are one standard deviation on either side of the mean.
5. Normal distributions are symmetric, but they do not have to be bell shaped.

Answer 1

How formal of a proof are you looking for?

Here is some intuition

If we look at the typical bell curve, the mean is the point the point in the middle. This point, by symmetry, corresponds to the point where 50% of the mass is to the left and 50% is to the right, so the mean is equal to median. It also corresponds to the most frequent (highest point on the curve) so it is equal to the mode.

For a more mathematical look, we can examine the kernel of the normal distribution given by

-(x-mu)^2.

By the properties of the square, this value is the same for -x as well as x. So it is symmetrical about mu.

Lastly, I'm not sure what the question is really asking for "bell-shaped" but if you take the sd to be large enough you can get shapes that are not at all "bell".

Hope this helps, let me know if you want more details.