Complete Business Statistics-Chapter 3

Complete Business Statistics Aczel Sounderpandian Chapter 3 Random Variables

• A random variable is a uncertain quantity whose value depends upon chance • A random variable is a function of the sample space • Discrete Random Variable – A discrete random variable can assume at most a countable number of values

• Continuous Random Variable – A continuous random variable may take on any value in an interval of numbers (i.e., its possible values are infinite)

• If the outcome of a trial can only be either a success or a failure, then the trial is a Bernoulli trial • The number of successes X in one Bernoulli trial, which can be 1 or 0, is a Bernoulli random variable

• Binomial Random Variable – An X that counts the number of successes in many independent, identical Bernoulli trials is called a binomial random variable

• Conditions for a binomial random variable – – –

The trials must be Bernoulli trials in that the outcomes can only be either success or failure The outcomes of the trials must be independent The probability of success in each trial must be constant

• Binomial Distribution – The probability mass function is given by

• n is the number of trials and p is the probability of success in each trial • In Binomial Distribution, the number of trials is fixed, however the number of successes desired is random. In Negative Binomial Distribution, the number of successes desired is fixed and the number of trials is random

• Negative Binomial Distribution – The probability mass function is given by

• Geometric Distribution – It is a special case of the negative binomial distribution where s=1 – The probability mass function is given by

• Hypergeomteric Distribution – When a pool of size N contains S successes and (N-S) failures, and a random sample of size n is drawn from the pool, the number of successes X in the sample follows a hypergeometric distribution

• Poisson Distibution – Poisson distribution may be used as a special case of the binomial distribution, where n may be very large and p may be very small, yet the product np lies between 0.01 to 50 – In such a scenario, the binomial formula may be approximated by

• The Poisson Distribution can be summarized as – The probability mass function is given by

• Continuous Random Variable – A continuous random variable is a random variable that can take on any value in an interval of numbers

• The probabilities associated with a continuous random variable X are determined by the probability density function of the random variable. The function, denoted f(x), has the following properties – f(x) >=0 for all x. – The probability that X will be between two numbers a and b is equal to the area under f(x) between a and b. – The total area under the entire curve of f(x) is equal to 1.00.

• The cumulative distribution function of a continuous random variable – F(x) =P(X