Continuous probability distribution.md (1187B)
1 +++ 2 title = 'Continuous probability distribution' 3 template = 'page-math.html' 4 +++ 5 # Continuous probability distribution 6 7 ## Normal distribution 8 9 Notation: 10 $X \sim N(\mu, \sigma^{2})$ 11 12 Percentile rules: 13 - 68%: within one standard deviation from mean 14 - 95%: within two standard deviations from mean 15 - 99.7%: within three standard deviations from mean 16 17 To find P(X ≤ x): 18 1. Find z score for x: $z = \frac{x - \mu}{\sigma}$ 19 2. Look up the cumulative probability for z. 20 3. P(X ≤ x) = P(Z ≤ z). So that’s your answer. 21 22 Z scores come from distribution $Z \sim N(0,1)$ 23 24 Also: P(X > x) = 1 - P(X ≤ x) 25 26 ### Central limit theorem 27 28 If you take sample size n ≥ 30, sample mean has approx normal distribution: 29 30 $\bar{X} \sim N(\mu, \frac{\sigma}{\sqrt{n}})$ 31 32 useful sometimes: $\frac{\sigma}{\sqrt{n}} = \sqrt{\frac{\sigma^{2}}{n}}$ 33 34 If the population is already normally distributed, the sample is always normally distributed for any n. 35 36 ### How do you know if something is normal? 37 38 Use a QQ plot. Put sample quantiles on y axis, theoretical quantiles on x axis. 39 If there’s a linear correlation, sample is normal. 40 In general, you can use QQ plots to compare two distributions/samples.