Probability Distributions

A probability distribution describes how the values of a random variable are distributed. It assigns probabilities to the possible values of the random variable, indicating the likelihood of each value occurring.

Discrete Probability Distributions

Discrete distributions deal with random variables that take on specific, distinct values. They are characterized by Probability Mass Functions (PMFs) which assign probabilities to discrete outcomes. They satisfy the following properties:

$$\begin{gather} 0 \le \mathbb{P}(x) \le 1 \qquad \forall x \\ \mathbb{P}(x=y)=\mathbb{P}(y) \qquad \forall y \\ \sum_X \mathbb{P}(x) = 1 \end{gather}$$

Examples of Discrete Distributions

• Bernoulli Distribution: The Bernoulli distribution describes experiments that have binary outcomes, e.g., result in either a success or a failure.

  • Example: Flipping a coin once, where heads is considered a success.

  • Formula:

    $$\mathbb{P}(x = k) = p^k (1-p)^{1-k} \quad \forall k \in \{0,1\}$$

    Where $p$ is the probability of success, $1-p$ is the probability of failure, and $k$ can be either 0 (failure) or 1 (success).

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• Poisson Distribution: The Poisson distribution describes the number of events occurring within a fixed interval of time or space when these events happen independently of each other and at a constant rate.

  • Example: The number of emails you receive in an hour.

  • Formula:

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

    Where $\lambda$ is the average number of events in the interval, $k$ is the number of occurrences, and $e$ is the Euler's constant.

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Continuous Probability Distributions

Continuous distributions describe random variables that can take on any value within a range. They are characterized by Probability Density Functions (PDFs), satisfying:

$$\begin{gather} 0 \le f(x) \\ \mathbb{P}(a \le x \le b) = \int_a^b f(x) \,dx \\ \int_\infty^{-\infty} f(x) \,dx = 1 \end{gather}$$

Examples of Continuous Distributions

• Uniform Distribution: The uniform distribution describes an equal probability for all values within a specified range.

  • Example: The probability of randomly selecting a number between 1 and 10.

  • Formula:

    $$f(x) = \begin{cases} \frac{1}{b-a} & \text{for } a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}$$

    Where $a$ and $b$ are the minimum and maximum values of the range.

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• Normal Distribution: The normal distribution, often referred to as the Gaussian distribution or bell curve, describes data that clusters around a mean.

  • Example: Heights of people in a population.

  • Formula:

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

    Where $\mu$ is the mean and $\sigma$ is the standard deviation.

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• Student's t-Distribution: The Student's t-distribution is used to estimate population parameters when the sample size is small and the population standard deviation is unknown.

  • Example: The distribution of the sample mean for small sample sizes.

  • Formula:

    $$f(x) = \frac{\Gamma \left(\frac{\nu+1}{2}\right)}{\sqrt{\nu \pi} \Gamma \left(\frac{\nu}{2}\right)} \left(1 + \frac{x^2}{\nu}\right)^{-\frac{\nu+1}{2}}$$

    Where $\nu$ is the degrees of freedom, and $\Gamma$ the gamma function.

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