Complete Business Statistics-Chapter 3

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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


• Geometric Distribution

– It is a special case of the negative binomial distribution where s=1




• 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


• 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 <= x) area under f(x) between the smallest possible value of X

• Uniform Distribution

– The probability density function is given

– Example: Elevator or Shuttle Bus Service



• Exponential Distribution

– Continuous limit of geometric distribution

– If an event occurs with an average frequency of λ occurrences per hour and this average frequency

is constant in that the probability that the event will occur during any tiny duration t is λt. Suppose

further we arrive at the scene at any giventime and wait till the event occurs. The waiting time will

then follow an exponential distribution – Examples of exponential distribution

• The time between two successive breakdowns of a machine will be exponentially distributed. This information is relevant to

maintenance engineers. The mean μ in this case is known as the mean time between failures, or MTBF

• The life of a product that fails by accident rather than by wear-and-tear follows an exponential distribution. Electronic

components are good examples. This information is relevant to warranty policies

• The time gap between two successive arrivals to a waiting line, known as the inter-arrival time, will be exponentially

distributed. This information is relevant to waiting line management

– The probability distribution function is given by

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Complete Business Statistics-Chapter 3