Probability Density Function

A random variable $X$ has a normal distribution $X\sim N(\mu ,{\sigma }^2)$ if and only if if has a pdf in the form of

for $-\mathrm{\infty }<x<\mathrm{\infty }$ where ${\sigma }^2>0$

If $X\sim N(\mu ,{\sigma }^2)$ then

It follows that

Moment-generating function

And for the standard normal distribution $Z\sim N(0,1)$:

If $X\sim Bin(n,p)$ then the moment-generating function of

approaches that of the standard normal distribution

then

(sampling distribution of the Binomial Distribution goes to normal :D)

Standard Normal Distribution

• Standard Normal Distribution (SND)
  • a normal distribution with $\mu =0,\sigma =1({\sigma }^2=1)$
  • represent random variables that have the SND with a Z
    • $Z\sim N(0,1)$
  • standardizing normally distributed random variables
    • converting it to the SND
    • $X\sim N(\mu ,{\sigma }^2)$
    • $Z=\frac{X-\mu }{\sigma }$
    • mean - mean = 0, sd/sd = 1
    • example
      • $P(X<0.22)⟶P(\frac{x-\mu }{\sigma }<\frac{0.22-\mu }{sigma})⟶P(Z<-1.5)$
  • PDF
    • 
  • CDF
    • $P(Z\le -1.31)$
    • 
• PDF
  • $$f(x)=\frac{1}{\sqrt{2\pi }\sigma }{e}^{-\frac{1}{2{\sigma }^2}(x-\mu {)}^2}$$
  • for $-\mathrm{\infty }<x<\mathrm{\infty }$
  • 2 parameters = $\mu ,\sigma /{\sigma }^2$
  • if x is normally distributed with mean $\mu$ and variance ${\sigma }^2$, we write it as $X\sim N(\mu ,{\sigma }^2)$
• symmetric about $\mu$
  • $\mu$ is the mean and the median
  • empirical rule
    • 
• many techniques are based on it
• real world variables, bell shaped curved
  Z distribution