$X$ is a Gaussian random variable with $\nu_x=0$ and $\sigma_X$. Plot $P(X>1)$ as a function of $\sigma_X$

Question

So, the pmf of Gaussian is:

$f_X(x)=\frac{1}{(2\pi\sigma_x^2)^{0.5}}e^{\frac{-(x-\nu_x)^2}{2\sigma_x^2}}$ where $\nu_x$ is the mean. As a result we get: $f_X(x)=\frac{1}{(2\pi\sigma_x^2)^{0.5}}e^{\frac{-x^2}{2\sigma_x^2}}$ So, $P(x>1)=\int_{-\infty}^{\infty} f_X(x)dx=\int_{-\infty}^{\infty}\frac{1}{(2\pi\sigma_x^2)^{0.5}}e^{\frac{-(x-\nu_x)^2}{2\sigma_x^2}}dx$ By replacement of variables $z=\frac{x}{\sigma_x}$ I get: $P(z>\frac{1}{\sigma_x})=\int_{\frac{1}{\sigma_x}}^{\infty}\frac{1}{(2\pi)^{0.5}}e^{\frac{-z^2}{2}}dz$$=Q(\frac{1}{\sigma_x})$. Assuming I got this right how would I plot this?

Answer 1

Here is a plot computed and made using R statistical software. Without software you will need to standardize and consult tables of the standard normal CDF at enough values to make a plot by hand. Note that pnorm is a normal CDF in R.

I assume $X \sim Norm(mean= 0, sd = \sigma),$ where $\sigma > 0.$ Define $g(\sigma) = P(X > 1).$ (If the stated task is to plot $g$ against $\sigma,$ I wonder if a plot against $1/\sigma$ is acceptable. What do you think you are supposed to learn from doing this?)

 sg = seq(.001, 25, by = .001)   # many POSITIVE values of sg, between 0 and 25
 g = 1 - pnorm(1, 0, sg)                   
 plot(sg, g, lwd=2, type="l", col="blue")
 abline(v=0, col="green2")       # y-axis
 abline(h=0, col="green2")       # x-axis

Notice the "flat" region near 0, until $\sigma$ gets large enough so there is appreciable probability above 1. (Also, do you think the curve approaches .5 for sufficiently large $\sigma$?)

Zooming in on the "flat spot":

 sg = seq(.0001, 2, by = .0001)      
 g = 1 - pnorm(1, 0, sg)                   
 plot(sg, g, lwd=2, type="l", col="blue")
 abline(v=0, col="green2")          # y-axis
 abline(h=0, col="green2")          # x-axis