# Inequalities in Normal Distributions

1. Sep 21, 2015

### Cpt Qwark

1. The problem statement, all variables and given/known data

How does P(-1<Z<1) equal to 1-2P(Z>1)?
(So you can find the values on the Normal Distribution Table)
2. Relevant equations

3. The attempt at a solution
I tried P(-1+1<Z+1<1+1) but ended up with P(1<Z+1<2).

2. Sep 21, 2015

### ehild

The normal distribution function is symmetric, f(z)=f(-z). The table shows the values of Cumulated Distribution Function F(Z) =P(-∞<z <Z). The probability P(Z<z<∞)=1-F(Z). Because of the symmetry, P(Z<z<∞)=P(-∞<z<-Z), F(-Z)=1-F(Z). The probability that the variable z is between -Z and Z is $$P(-Z<z<Z)=F(Z)-F(-Z) = F(Z)-(1-F(Z))$$.

3. Sep 21, 2015

### SteamKing

Staff Emeritus
The standard normal curve is symmetrical about a mean value μ = 0 like this:

http://www.spiritsd.ca/curr_content/mathb30/data/les6/images/norm_percent.gif [Broken] ​

The normal curve is scaled such that the total area under the curve is 1.

Since you are trying to find P(-1 < Z < 1), don't you see how that's the same probability as 1 - 2P(Z>1)?

Last edited by a moderator: May 7, 2017