What is probability distribution?

• A function of a discrete variable whose integral over any interval is the probability that the random variable specified by it will lie within that interval.

• It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).

• A statistical function that describes possible values and likelihoods that a random variable can take within a given range.

Continuous Distributions

Normal Distribution symmetric about the mean. It follows a bell-shaped curve. Widely used in statistics and natural phenomena. Example: Heights of people, test scores, etc. Exponential Distribution used to model the time between events in a Poisson process. Example: Time between arrivals of buses, time between radioactive decays.

Discrete Distributions

Binomial Distribution models the number of successes in a fixed number of independent Bernoulli trials. Example: Flipping a coin a fixed number of times and counting the number of heads. Poisson Distribution expresses the probability of a number of events occurring within a fixed interval of time or space. Example: The number of emails received in an hour, number of accidents at an intersection in a day.

There are others and each have different characteristics and applications.

The PMF gives the probability that a discrete random variable takes a particular value.

For a discrete random variable \(X\): \[ P(X = x) = f(x) \]

For continuous random variables, the probability density function (PDF) represents the likelihood of a value within a range.

The area under the curve for a given interval \([a, b]\) gives the probability:

\[ P(a \leq X \leq b) = \int_a^b f(x) \, dx \]

library(ggplot2)

x <- seq(-5, 5, by = 0.1)
y <- dnorm(x)

ggplot(data.frame(x, y), aes(x, y)) +
  geom_line(color = "blue") +
  ggtitle("Normal Distribution Curve") +
  xlab("X") +
  ylab("Density") +
  theme_minimal()