Discrete Probability Distributions

Discrete Probability Distributions

A discrete probability distribution is a probability distribution that describes the likelihood of outcomes for a discrete random variable. A discrete random variable is one that can take on a finite or countably infinite number of possible values. For example, the number of heads in a series of coin tosses or the number of cars passing through a toll booth in an hour are discrete random variables.

A probability distribution specifies the probability of each possible value of the random variable. In the case of a discrete random variable, the probability distribution is typically represented in one of two ways: a probability mass function (PMF) or a probability table.

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Key Properties of Discrete Probability Distributions

1. The Random Variable Takes Discrete Values:

   • A discrete random variable $X$ can take a finite or countably infinite number of values. For example, $X$ could represent the number of heads in 10 coin tosses, which can take any integer value from 0 to 10.

2. The Sum of Probabilities is 1:

   • The total probability for all possible outcomes must sum to 1. This is a fundamental property of any probability distribution, meaning that one of the outcomes must occur.
   $$\sum_{x} P(X = x) = 1$$

   where $x$ represents all possible values the random variable $X$ can take.

3. Non-Negative Probabilities:

   • The probability of each possible outcome is always a non-negative number, meaning that $P(X = x) \geq 0$ for all possible values $x$ of the random variable.

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Common Types of Discrete Probability Distributions

There are several commonly used discrete probability distributions in statistics and probability theory:

1. Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials (experiments with two outcomes, often called "success" and "failure"). It is defined by two parameters:

• $n$: the number of trials,
• $p$: the probability of success on a single trial.

The probability mass function (PMF) for the binomial distribution is given by:

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

Where:

• $\binom{n}{k}$ is the binomial coefficient, representing the number of ways to choose $k$ successes from $n$ trials.
• $p^k$ is the probability of having exactly $k$ successes.
• $(1-p)^{n-k}$ is the probability of having $n-k$ failures.

Example: If you flip a fair coin 5 times, the number of heads $X$ follows a binomial distribution with parameters $n = 5$ and $p = 0.5$. The probability of getting exactly 3 heads is:

$$P(X = 3) = \binom{5}{3} (0.5)^3 (0.5)^2 = 10 \times 0.125 \times 0.25 = 0.3125$$

2. Poisson Distribution

The Poisson distribution models the number of events occurring in a fixed interval of time or space, where the events occur independently and at a constant average rate. The distribution is defined by a single parameter $\lambda$, which is the average number of events in the interval.

The probability mass function for the Poisson distribution is:

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

Where:

• $k$ is the number of events (can be 0, 1, 2, …),
• $\lambda$ is the average rate of occurrence (mean number of events),
• $e$ is Euler's number (approximately 2.71828).

Example: If the average number of cars passing through a toll booth is 3 per hour ($\lambda = 3$), the probability of exactly 4 cars passing in one hour is:

$$P(X = 4) = \frac{3^4 e^{-3}}{4!} = \frac{81 e^{-3}}{24} \approx 0.168$$

3. Geometric Distribution

The geometric distribution models the number of trials required to get the first success in a sequence of independent Bernoulli trials. It is defined by the probability of success on each trial, $p$.

The probability mass function is:

$$P(X = k) = (1 - p)^{k-1} p$$

Where:

• $k$ is the number of trials until the first success (can be 1, 2, 3, …),
• $p$ is the probability of success on each trial.

Example: If the probability of getting heads on a coin flip is $p = 0.5$, the probability that the first heads appears on the 3rd flip is:

$$P(X = 3) = (1 - 0.5)^{2} \times 0.5 = 0.25 \times 0.5 = 0.125$$

4. Hypergeometric Distribution

The hypergeometric distribution is similar to the binomial distribution, but it applies when sampling is done without replacement. It models the probability of $k$ successes in a sample of size $n$ drawn from a population of size $N$ containing $K$ successes.

The probability mass function is:

$$P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}$$

Where:

• $N$ is the total population size,
• $K$ is the number of successes in the population,
• $n$ is the sample size,
• $k$ is the number of successes in the sample.

Example: In a population of 20 people, 8 are students. If 5 people are selected randomly without replacement, the probability that 3 of the selected people are students is:

$$P(X = 3) = \frac{\binom{8}{3} \binom{12}{2}}{\binom{20}{5}} = \frac{56 \times 66}{15504} \approx 0.239$$

5. Negative Binomial Distribution

The negative binomial distribution generalizes the geometric distribution. It models the number of trials needed to achieve a fixed number of successes. It is defined by two parameters:

• $r$: the number of successes,
• $p$: the probability of success on each trial.

The probability mass function is:

$$P(X = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}$$

Where $k$ is the total number of trials needed to achieve $r$ successes.

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Expectation and Variance of Discrete Distributions

1. Expectation (Mean): The expectation (or mean) of a discrete random variable $X$ is the weighted average of all possible values of $X$, weighted by their probabilities:

   $$E[X] = \sum_{x} x P(X = x)$$

2. Variance: The variance of $X$ measures the spread or dispersion of the distribution, and is given by:

   $$\text{Var}(X) = E[(X - \mu)^2] = \sum_{x} (x - \mu)^2 P(X = x)$$

   Where $\mu = E[X]$ is the mean.

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Summary

• Discrete probability distributions describe the probability of outcomes for discrete random variables, and they can be represented by a probability mass function (PMF).
• Common discrete distributions include the binomial, Poisson, geometric, hypergeometric, and negative binomial distributions.
• These distributions have applications in various fields, including quality control, reliability testing, and modeling rare events.
• The expectation and variance of a discrete random variable can be computed from the probability distribution, providing important measures of central tendency and dispersion.