Some Common Distributions

Uniform distribution for discrete random variable

A uniform distribution for a discrete random variable signifies a scenario where each possible outcome has an equal probability of occurring. This implies that the probability mass function (PMF) assigns the same value to each outcome within a finite set.

For instance, consider rolling a fair six-sided die; if each face has an equal chance of appearing, this would lead to a uniform distribution among the six possible outcomes.

Mathematically, if X represents the discrete random variable, and the outcomes are

$x₁, x₂, ..., xₙ$, then the PMF is given by the following formula

$P(X = xᵢ) = \frac{1}{n} \ for\ all \ i$,

where n is the total number of possible outcomes.

Uniform distribution for continuous random variable

A uniform distribution for a continuous random variable describes a situation where the probability density function (PDF) remains constant within a specified interval. The PDF of a continuous uniform distribution is constant, indicating a constant probability density across the interval. Mathematically, if $X$is a continuous random variable distributed uniformly over the interval $[a, b]$, then the probability density function is

$f(x) = \frac{1}{(b - a)} \ for \ a ≤ x ≤ b.$

Binomial distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success, denoted by p. Commonly used in scenarios such as coin flips or product defect rates, the binomial distribution's probability mass function (PMF) calculates the likelihood of observing a specific number of successes in a fixed number of trials. The key parameters are the number of trials $(n)$ and the probability of success $(p)$on each trial.

Mathematically, if X represents the number of successes in n trials, then the probability mass function is:

$P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$,

where $\binom{n}{k}$ denotes the binomial coefficient.

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Normal distribution

The normal distribution, also known as the Gaussian distribution, is the most common distribution in statistics. The normal distribution is characterized by a symmetric bell-shaped curve. In a normal distribution, the distribution is entirely defined by its mean (μ) and standard deviation (σ). Many natural phenomena, tend to follow a normal distribution.

Its probability density function is described by the formula

$f(x) = \frac{1}{\sigma \sqrt{2 \pi}} \, e^{-\frac{(x - \mu)^2}{2 \sigma^2}}$

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Student's t-Distribution

The Student's t-distribution (often referred to simply as the t-distribution) is a probability distribution that arises in statistical inference when the sample size is small or when the population variance is unknown. It is widely used in hypothesis testing, confidence interval estimation, and linear regression analysis. The shape of the t-distribution depends on a parameter known as degrees of freedom, which is related to the sample size. As the degrees of freedom increase, the t-distribution approaches the standard normal distribution. However, for smaller sample sizes, the t-distribution has heavier tails, making it more robust against outliers compared to the normal distribution. Understanding the t-distribution is crucial for conducting accurate statistical analyses, especially when dealing with small samples or unknown population variances.

Poisson Distribution

The Poisson distribution is employed to model the number of events occurring within a fixed interval of time (or space), given a known average rate of occurrence. It is particularly useful in scenarios where events happen independently and at a constant average rate, such as the number of customer arrivals in a queue or the number of emails received per day. The Poisson distribution's probability mass function calculates the likelihood of observing a specific number of events in a given interval. Its key parameter is lambda, representing the average rate of occurrence. Mathematically, if X represents the number of events occurring in a fixed interval, then the probability mass function is:

$P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$

where k is a non-negative integer.