# area under curve must be 1, function intersects with y-axis above 0

what is the best way to find a function that looks like a normal distribution, when the curve intersects the y-axis above 0 (say 0.3) and the area must be 1 (100%)?

https://i.stack.imgur.com/5cJSy.jpg

## 1 Answer

The PDF of the normal distribution is $$\frac 1 {\sqrt{2 \pi \sigma^2}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$$. If $$\mu=0$$, then the PDF at zero is $$\frac 1 {\sqrt{2 \pi \sigma^2}}$$. So if you want the y-intercept to be $$y_0$$, you just need to solve $$y_0=\frac 1 {\sqrt{2 \pi \sigma^2}}$$ for $$\sigma$$. That gives $$\frac 1{y_0^2}=2\pi\sigma^2\rightarrow\sigma^2=\frac1{{2\pi y_0^2}}\rightarrow \sigma = \frac{1}{y_0 \sqrt{2\pi}}$$ for $$y_0>0$$.

• Thank you very much. I'm making a thinking error. I need to express that the probability of spending 0 (x) is 0.3 (y), and the probability of spending less than 0 (no negative numbers on the x-axis) is 0 (y). from there I will find the CDF. – Phi Al May 9 at 9:58