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Discrete and continuous probability modelsPresentation Transcript
Discrete and Continuous
Akshay Kr Mishra-100106039
Sharda University, 4th yr ;ME
• A probability distribution is a mathematical
model that relates the value of the variable
with the probability of occurrence of that value
in the population.
• There are 2 types of probability Distribution-
1. Continuous Probability Distribution
2. Discrete Probability Distribution.
When the variable being
measured is expressed on a
continuous scale, its
probability distribution is
called a continuous
distribution. Ex- The
probability distribution of
metal layer thickness is
Discrete distributions. When
the parameter being
measured can only take on
certain values, such as the
integers 0, 1 etc. the
probability distribution is
called a discrete
Ex- distribution of the number
of nonconformities or
defects in printed circuit
boards would be a discrete
Some Imp. Terms
• Mean- The Mean of a
probability distribution is a
measure of the central
tendency in the distribution,
or its location.
• Variance- The scatter, spread,
or variability in a distribution
is expressed by the variance.
• Standard Deviation- The
standard deviation is a
measure of spread or scatter
in the population expressed
in the original terms.
Types Of Discrete Distribution
• Hyper geometric Distribution- An
appropriate probability model for selecting a
random sample of n items without replacement
from a lot of N items of which D are
nonconforming or defective.
• In these applications, x usually is the class of
interest and then that x is the hyper geometric
• Binomial Distribution- Lets consider a process
of ‘n’ independent trials.
• When the outcome of each trial is either a
“success” or a “failure,” the trials are called
• If the probability of “success” on any trial—say,
p—is constant, then the number of “successes” x
in n Bernoulli trials has the binomial
• The binomial distribution is used frequently in
• It is the appropriate probability model for
sampling from an infinitely large population,
where p represents the fraction of defective or
nonconforming items in the population.
• In these applications, x usually represents the
number of nonconforming items found in a
random sample of n items.
• We note a Important
fact here and that is the
mean and variance of
the Poisson distribution
are both equal to the
• A typical application of the Poisson distribution in
quality control is as a model of the number of defects
or nonconformities that occur in a unit of product.
• In fact, any random phenomenon that occurs on a per
unit (or per unit area, per unit volume, per unit time,
etc.) basis is often well approximated by the Poisson
• It is possible to derive the Poisson distribution as a
limiting form of the binomial distribution.
• That is, in a binomial distribution with parameters n
and p, if we let n approach infinity and p approach zero
in such a way that np = lambda is a constant, then the
Poisson distribution results.
Types of Continuous Distribution
• Lognormal Distribution-
• The lifetime of a product that degrades over time is
often modelled by a lognormal random variable. For
example-the lifetime of a semiconductor laser.
• However, because the lognormal distribution is
derived from a simple exponential function of a
normal random variable, it is easy to understand and
easy to evaluate probabilities.
• Normal Distribution- The normal distribution
is probably the most important distribution in
both the theory and application of statistics.
• If x is a normal random variable, then the
probability distribution of x is defined as
• The normal distribution is used so much that we
frequently employ a special notation, to
imply that x is normally distributed with mean and
• The visual appearance of the normal distribution is
a symmetric, unimodal or bell-shaped curve.
Area Under Normal Distribution.
• The Normal Distribution has many useful properties and
one which has found its world wide use is the “Central
• Central Limit Theorem- The central limit theorem implies
that the sum of n independently distributed random
variables is approximately normal, regardless of the
distributions of the individual variables.
• The approximation improves as n increases.
• Exponential Distribution-
Area Under the Exponential Distribution
The exponential distribution is widely used in the field of reliability
engineering as a model of the time to failure of a component or
In these applications, the parameter is called the failure rate of the
system, and the mean of the distribution is called the
mean time to failure.