Introduction to Discrete & Continuous Probability Distributions

 

✅ 1. What is a Probability Distribution?

A probability distribution describes how probabilities are distributed over the values of a random variable.

• Random Variable: A variable whose values are outcomes of a random phenomenon.

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🧮 2. Types of Probability Distributions

Type	Description	GIS Example
Discrete	Takes countable values	Number of landslides per year in a          valley
Continuous	Takes infinite values over an                 interval	Rainfall (mm), elevation, temperature

📌 Discrete Probability Distributions

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🎯 3. Binomial Distribution

✅ Definition:

Used when an experiment is repeated n times, and each trial has two outcomes: success or failure.

✅ Conditions:

1. Fixed number of trials (n)

2. Only two possible outcomes per trial (success/failure)

3. Constant probability of success (p)

4. Trials are independent

✅ Formula:

$$P(X=k)=\left(\genfrac{}{}{0px}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$

Where:

• $\mathrm{<span\; style="font-size:\; medium;">n:\; number\; of\; trials</span>}$

• $\mathrm{<span\; style="font-size:\; medium;">k:\; number\; of\; successes</span>}$

• $p$: probability of success

• $\binom{n}{k}$: combination formula

📍 Example:

Suppose there’s a 0.3 chance that a village is at high flood risk. Out of 5 randomly selected villages, what is the probability exactly 2 are high risk?

$$n = 5, \quad k = 2, \quad p = 0.3$$
$$P(X = 2) = \binom{5}{2} (0.3)^2 (0.7)^3 = 10 \cdot 0.09 \cdot 0.343 = 0.3087$$

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🔢 4. Poisson Distribution

✅ Definition:

Used to model the number of times an event occurs in a fixed interval of time or space, where events occur independently.

✅ Conditions:

1. Events occur one at a time

2. Average rate $\lambda$ is constant

3. Events are independent

✅ Formula:

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

Where:

• $\lambda$: average number of occurrences

• $k$: actual number of occurrences

📍 Example:

On average, 3 landslides occur in a region each year. What is the probability there will be exactly 2 landslides in the coming year?

$$\lambda = 3, \quad k = 2$$
$$P(X = 2) = \frac{e^{-3} \cdot 3^2}{2!} = \frac{e^{-3} \cdot 9}{2} \approx 0.224$$

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📊 Continuous Probability Distributions

🌐 5. Normal Distribution

✅ Definition:

A continuous distribution that is symmetric and bell-shaped, representing many natural and measurement-based phenomena.

✅ Conditions:

1. Data is continuous

2. Symmetric around the mean

3. Mean = Median = Mode

4. Follows the 68-95-99.7 rule

✅ Formula (Probability Density Function):

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

Where:

• $\mu$: mean

• $\sigma$: standard deviation

📍 Example:

The average elevation of a region is 2,000 meters with a standard deviation of 200 meters.
We want to know the probability that a randomly chosen location has elevation between 1,800 and 2,200 meters.

This range is within ±1 standard deviation, so by the empirical rule,

$$P(1800 \leq X \leq 2200) \approx 68\%$$

🎯 7. Applications in GIS

• Binomial: Probability a satellite detects high NDVI in certain % of land

• Poisson: Earthquake events in a seismic zone

• Normal: Distribution of temperatures or rainfall across regions