Discrete and Continuous Distributions

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.

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1. Discrete Distributions

Discrete distributions describe the probabilities of outcomes for a discrete random variable, which takes on a countable number of values.

Characteristics:

• 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.

Common Discrete Distributions:

1.1. Bernoulli Distribution

• Describes a random experiment with two outcomes: success ($1$) and failure ($0$).
• Example: Tossing a coin (success = heads, failure = tails).

• 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.

1.2. Binomial Distribution

• Describes the number of successes in $n$ independent Bernoulli trials.
• Example: Number of heads in 10 coin tosses.

• PMF:

  \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \dots, n\]

  Where:

  • $n$: Number of trials
  • $p$: Probability of success

1.3. Poisson Distribution

• Models the number of events in a fixed interval of time or space.
• Example: Number of customer arrivals at a store in an hour.

• PMF:

  \[P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k = 0, 1, 2, \dots\]

  Where $\lambda$ is the average rate of occurrences.

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2. Continuous Distributions

Continuous distributions describe probabilities for a continuous random variable, which can take on an infinite number of values within a range.

Characteristics:

• 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.

Common Continuous Distributions:

2.1. Uniform Distribution

• All outcomes in a range $[a, b]$ are equally likely.
• Example: Randomly selecting a number between 0 and 10.

• PDF:

  \[f(x) = \frac{1}{b-a}, \quad a \leq x \leq b\]

2.2. Normal (Gaussian) Distribution

• Describes data that clusters around a mean ($\mu$), with a spread determined by the standard deviation ($\sigma$).
• Example: Heights of people in a population.

• PDF:

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

2.3. Exponential Distribution

• Models the time between events in a Poisson process.
• Example: Time until the next customer arrives.

• PDF:

  \[f(x) = \lambda e^{-\lambda x}, \quad x \geq 0\]

  Where $\lambda$ is the rate parameter.

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Key Differences Between Discrete and Continuous Distributions

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)

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Applications in Real Life

Discrete Distributions:

1. Binomial: Probability of a specific number of defective items in a batch.
2. Poisson: Modeling call arrivals at a call center.

Continuous Distributions:

1. Normal: Analyzing test scores in a standardized exam.
2. Exponential: Estimating the time between server requests.

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Visualization

Discrete:

• Probability mass function represented as a bar graph.

Continuous:

• Probability density function represented as a smooth curve.

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Conclusion

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.

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Next Steps: Sampling Techniques