Probability distributions describe how the values of a random variable are distributed. These distributions are categorized into discrete and continuous, depending on the nature of the random variable.
Discrete distributions describe the probabilities of outcomes for a discrete random variable, which takes on a countable number of values.
- The probabilities of all possible outcomes add up to 1.
- Examples of discrete random variables: Number of heads in coin tosses, number of customers in a queue.
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Describes a random experiment with two outcomes: success (
$1$ ) and failure ($0$ ). -
Example: Tossing a coin (success = heads, failure = tails).
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Probability Mass Function (PMF):
$$P(X = x) = p^x (1-p)^{1-x}, \quad x \in {0, 1}$$ Where
$p$ is the probability of success.
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Describes the number of successes in
$n$ independent Bernoulli trials. -
Example: Number of heads in 10 coin tosses.
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PMF:
$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \dots, n$$ Where:
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$n$ : Number of trials -
$p$ : Probability of success
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Models the number of events in a fixed interval of time or space.
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Example: Number of customer arrivals at a store in an hour.
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PMF:
$$P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k = 0, 1, 2, \dots$$ Where
$\lambda$ is the average rate of occurrences.
Continuous distributions describe probabilities for a continuous random variable, which can take on an infinite number of values within a range.
- Probabilities are represented as areas under a probability density function (PDF).
- The total area under the PDF equals 1.
- Examples of continuous random variables: Height, weight, temperature.
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All outcomes in a range
$[a, b]$ are equally likely. -
Example: Randomly selecting a number between 0 and 10.
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PDF:
$$f(x) = \frac{1}{b-a}, \quad a \leq x \leq b$$
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Describes data that clusters around a mean (
$\mu$ ), with a spread determined by the standard deviation ($\sigma$ ). -
Example: Heights of people in a population.
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PDF:
$$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$
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Models the time between events in a Poisson process.
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Example: Time until the next customer arrives.
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PDF:
$$f(x) = \lambda e^{-\lambda x}, \quad x \geq 0$$ Where
$\lambda$ is the rate parameter.
| Feature | Discrete Distribution | Continuous Distribution |
|---|---|---|
| Random Variable | Countable values (e.g., 0, 1, 2) | Infinite values within a range |
| Representation | PMF (Probability Mass Function) | PDF (Probability Density Function) |
| Examples | Binomial, Poisson | Normal, Uniform, Exponential |
| Probability | Exact value has non-zero probability | Exact value has zero probability (area under curve for range) |
- Binomial: Probability of a specific number of defective items in a batch.
- Poisson: Modeling call arrivals at a call center.
- Normal: Analyzing test scores in a standardized exam.
- Exponential: Estimating the time between server requests.
- Probability mass function represented as a bar graph.
- Probability density function represented as a smooth curve.
Understanding discrete and continuous distributions is essential for modeling real-world phenomena and conducting statistical analysis. By selecting the appropriate distribution, you can gain deeper insights into data and make informed predictions.
Next Steps: Sampling Techniques