This document discusses different types of probability distributions used in statistics. There are two main types: continuous and discrete distributions. Continuous distributions are used when variables are measured on a continuous scale, while discrete distributions are used when variables can only take certain values. Some important continuous distributions mentioned are the normal, lognormal, and exponential distributions. Important discrete distributions include the binomial, hypergeometric, and Poisson distributions. Key terms like mean, variance, and standard deviation are also defined. Examples are provided to illustrate how these probability distributions are applied in fields like quality control and reliability engineering.

1. Discrete and Continuous
Probability Models
Akshay Kr Mishra-100106039
Sharda University, 4th yr ;ME

2. Probability Distribution?
• 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.

3. Continuous distributions-
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
continuous.
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
distribution.
Ex- distribution of the number
of nonconformities or
defects in printed circuit
boards would be a discrete
distribution

4. 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.

5. 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
random variable.

7. • 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
Bernoulli trials.
• 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
distribution.

9. • The binomial distribution is used frequently in
quality engineering.
• 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.

10. Poisson’s Distribution
• We note a Important
fact here and that is the
mean and variance of
the Poisson distribution
are both equal to the
parameter Lambda.

11. • 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
distribution.
• 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.

12. Types of Continuous Distribution
• Lognormal Distribution-

13. • 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.

14. • 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
follows.

15. • The normal distribution is used so much that we
frequently employ a special notation, to
imply that x is normally distributed with mean and
variance.
• The visual appearance of the normal distribution is
a symmetric, unimodal or bell-shaped curve.
Area Under Normal Distribution.

16. • The Normal Distribution has many useful properties and
one which has found its world wide use is the “Central
Limit Theorem”.
• 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.

17. • Exponential Distribution-

18. 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
system.
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.

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