Normal distribution

Definition: $X\sim N(\mu ,{\sigma }^2)$

• Continuous random variable
• based on Standard normal distribution
• We say that $X$ is normal random variable, or simply that $X$ is normally distributed, with parameters $\mu$ and ${\sigma }^2$ denoted by $X\sim N(\mu ,{\sigma }^2)$
  • $X\sim N(5,10)$

-
  ```functionplot
  ---
  xLabel:
  yLabel:
  bounds: [-5,15,0,0.15]
  disableZoom: true
  grid: true
  ---
  f(x)=1/(sqrt(2*PI)*sqrt(10))* E^(-(x-5)^2/(2*10))
  ```

• Properties:
  • ${\int }_{-\infty }^{\infty }f(x)dx=1$
  • Expected value $E[X]=\mu$
  • Variance $Var(x)={\sigma }^2$
  • Graph of $f$ is symmetric in the line $x=\mu$
  • $f$ is maximized when $x=\mu$
  • $f$ has two Infection points at $x=\mu \pm \mu$
• Conditions
  • $f(x)\ge 0\forall x\in (-\infty ,\infty )$

Probability density function

• $f(x)=\frac{1}{\sqrt{2\pi }\sigma }{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}},\infty <x<\infty$

Normal approximation to Binomial distribution:

• Use The DeMoivre-Laplace limit theorem to evaluate a Binomial distribution when the number of trials become large.
• $X\sim B(n,p)\to X\sim N(np,npq)$ where:
  • $np\ge 5$
  • $nq=n(1-p)\ge 5$
• Continuity correction:
  • Convert from discrete integer valued (binomial) to continuous (normal)
  • $P(X=i)$ as $P(i-0.5<X<i+0.5)$
  • $P(X<i)\to P(X<i+0.5)$
  • $P(X\le i)\to P(X<-0.5)$
  • $P(X>i)\to P(X<i+0.5)$
  • $P(X\ge i)\to P(X<i-0.5)$

Sum of independent normal distribution:

• Sum of independent variables
• If $X_i\sim N({\mu }_i,{\sigma }_i^2)$, then $\sum _{i=1}^n{X}_i\sim N(\sum _{i=1}^n{\mu }_i,\sum _{i=1}^n{\sigma }_i^2)$