Distribution (Binomial,
Poisson and Normal)
Distribution
refers to the pattern of values that a random variable can take and the
likelihood of each value occurring. In statistics, several common probability
distributions are used to model different types of data. Here's an overview of
three important distributions: the binomial, Poisson, and normal distributions.
1. Binomial
Distribution:
The binomial
distribution models the number of successes (usually denoted as "x")
in a fixed number of independent Bernoulli trials. A Bernoulli trial is an
experiment with two possible outcomes, typically labeled as "success"
and "failure." The key characteristics of the binomial distribution
are:
- Each trial is
independent of the others.
- There are
only two possible outcomes in each trial.
- The
probability of success (p) remains constant across all trials.
The probability
mass function (PMF) of the binomial distribution is given by:
P(X = x) = C(n,
x) * p^x * (1 - p)^(n - x)
Where:
- C(n, x) is
the binomial coefficient, equal to n! / (x! * (n - x)!).
- n is the number
of trials.
- p is the
probability of success in each trial.
- X is the
random variable representing the number of successes.
The binomial
distribution is commonly used in scenarios where we want to calculate the
probability of getting a certain number of successes in a fixed number of
trials, such as coin tosses or the number of successes in a batch of defective
items.
2. Poisson
Distribution:
The Poisson
distribution models the number of events that occur within a fixed interval of
time or space when events happen at a constant rate and independently of the
time since the last event. The key characteristics of the Poisson distribution
are:
- Events occur
randomly and independently.
- The rate of
occurrence is constant over time.
The probability
mass function (PMF) of the Poisson distribution is given by:
P(X = x) = (λ^x
* e^(-λ)) / x!
Where:
- λ (lambda) is
the average rate of events per unit time or space.
- X is the
random variable representing the number of events.
The Poisson
distribution is commonly used to model rare events, such as the number of
arrivals at a service center in a given time period or the number of defects in
a product.
3. Normal
Distribution (Gaussian Distribution):
The normal
distribution is one of the most widely used probability distributions in
statistics. It describes continuous random variables that are symmetrically
distributed around their mean. The key characteristics of the normal
distribution are:
- It is
symmetric, bell-shaped, and unimodal.
- The mean,
median, and mode are all equal.
- The tails of
the distribution extend to infinity but never touch the x-axis.
The probability
density function (PDF) of the normal distribution is given by:
f(x) = (1 / (σ
* √(2π))) * e^(-(x - μ)^2 / (2 * σ^2))
Where:
- μ (mu) is the
mean of the distribution.
- σ (sigma) is
the standard deviation of the distribution.
- x is the
random variable.
The normal
distribution is commonly used in various statistical analyses and hypothesis
testing, as many natural phenomena and measurement errors tend to follow this
distribution. It is also essential in the Central Limit Theorem, which states
that the sample means of sufficiently large samples from any distribution will
follow a normal distribution, even if the population itself does not follow a
normal distribution.
Understanding
these fundamental distributions is crucial in various statistical analyses and
helps in selecting appropriate models to represent different types of data.
