Expected Value and Variances

There are 2 fundamental quantities of probability distributions: expected value and variance.

Expected value:

• The simplest and most useful summary of the distribution of a random variable is the “average” of the values it takes on.
• (Please see references for equation)

Variance : 

• The variance is a measure of how broadly distributed the r.v. tends to be.
• It’s defined as the expectation of the squared deviation from the mean:
  • Var(X) = E[(X − E(X))2 ]
• In general terms, it is the expected squared distance of a value from the mean.

 

Looking at different distributions presents an interesting take on these two quantities:

1. Bernoulli Distribution
2. Uniform Distribution
3. Geometric Distribution
4. Binomial Distribution
5. Normal Distribution
6. Hypergeometric Distribution
7. Poisson Distribution

 

References:

• http://idiom.ucsd.edu/~rlevy/teaching/fall2008/lign251/lectures/lecture_3.pdf
• http://www.dma.unifi.it/~modica/2009-10/an1/appendici-Cormen.pdf
• http://terras-altas.net.br/MA-2013/statistics/probability%20distribution%20functions/Examples%20of%20Bernouilli%20distribution.pdf
  • This link has nice examples of several distributions
•  http://people.umass.edu/biep540w/pdf/bernoulli.pdf
• http://www.math.uah.edu/stat/interval/Bernoulli.html

 

Video:

• https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-proportions/v/mean-and-variance-of-bernoulli-distribution-example
  • The 2 videos in this link discusses the mean and variance of a bernoulli distribution

Code:

• evaluator_ips