Some Probability Distribution

Expectation and Variance

• expectation is the mean value of a sequence of values.      
• E(x) = sum(1...n) / n
• variance is the average of squared difference from mean. 
• Var(x) = E(x - E(x))^2
• standard deviation is the square root of variance.              
• Sigma = sqrt(Var(x))
• using standard deviation, we know what is normal(standard), what is extra large or small.

Uniform Distribution

•  a and b are solvable given E(x) and V(x)

Normal (Gaussian) Distribution

 Binomial Distribution (Bernoulli Trials)

• a sequence of independent yes/no experiment, called Bernoulli experiment
• when n = 1, binomial distribution is Bernoulli distribution

• e.g. 
• toss a coin 100 times, what is the prob. the head occurs 30 times? (n = 100, k = 30, p = 1/2)
• 
  

Poisson Distribution

Geometric Distribution

• modelling the trials up to and including the first success (k = 1, 2, 3, ...)
• e.g.
• toss a fair coin until the first heads, E(x) = 1/p = 2, which means on average tossing 2 times we will get heads.

• modelling the number of failures until the first success (k = 0, 1, 2, 3, ...)
• e.g.
• assume the success prob. of something is p, the mean number of success is: (1-p)/p.

Reference

• https://en.wikipedia.org/wiki/Uniform_distribution_(continuous),
• https://en.wikipedia.org/wiki/Bernoulli_trial
• https://en.wikipedia.org/wiki/Geometric_distribution