Random Variable

• A random variable represents a possible outcome (usually numeric) from a random experiments&amp
  • Let X be random variable for a number of heads in 3 coin flips 
  • X represents any value from sample space {0, 1, 2 or 3)
• An observation is a realization of random variable
  • Let x be an observed number of heads in 3 coin tosses, eg x = 0

Bernuolli Trials:

• Random trial with two outcomes. Eg. Success or Failure. 
• Random variable X often coded as 0 (Failure) and 1 (Success)
• Bernuolli Trail has probability of success usually denoted p. Eg. P (Success) = p (x =1) = p 
• Accordingly, probability of failure (1 – p) is usually denoted q = 1 - p. 
• Eg. p (Failure) = 1 – p = q 
• Probability of Bernuolli’s Distribution is: 
• Where x can be zero or one
• The Binomial Distribution which consists of a fixed number of statistically independent Bernoulli trials

Binomial Distributing Function:

Poisson Random Variable:

Poisson random variable represents the number of independent events that occur randomly over unit of times 

Example:

• Number of calls to a call centre in an hour 
• Number of floods in a river in a year 
• Number of misprint in a page

Characteristic:

• Count number of times as event occur during a given unit of measurement (e.g. time, area). 
• Number of events that occur in one unit is independent of other units. 
• Probability that events occurs over given unit is identical for all units (constant rate) 
• Events occur randomly 
• Expected number of events (rate) in each unit is denoted by λ (lambda)

Poisson Distribution:

• Let x be Poisson random variable for number of independent events over unit of measurements. 
• To define probability distribution need to know: rate of unit occurrence per unit - λ .