Random Variables

Random variable

Numerical measurement of outcome of an experiment.

Probability distribution

Specifies possible values and their probabilities.

Z	0	…	…	…	…	Total
P(Z)	…	…	…	…	…	1

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Discrete Random Variable

Discrete random variable

Takes on a set of separate value {0, 1, 2, 3, …}. Probability distribution assigns a probability to each possible value of .

• Sum of the probabilities for all the possible values equals .

Mean of discrete variable

Mean of discrete random variable is denoted by :

Properties of the mean:

• Linear transformation . Let , where are known constants:
• If are random variables with means and are known constants
• If denote random variables identically distributed
  • Same probability distribution
  • Same mean

Variance of discrete variable

Variance of a discrete random variable is denoted by :

Properties of the variance:

• Linear transformation is a random variable with variance . Let , where are known constants:
• If are random variables with means and are known constants
• If denote random variables identically distributed
  • Same probability distribution
  • Same variance

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Continuous Random Variable

Continuous random variable

Has possible values that form an interval

Probability density function (pdf)

The curve that determines the probabilities of intervals. Specifies the probability distribution.

Mean of continuous variable

Mean of a continuous random variable which has pdf is denoted by Greek letter

Variance of continuous variable

Variance of continuous random variable which has pdf is denoted by Greek letter

Quantile or percentile

For a continuous random variable , the th quantile, is a value such that