Explain continuous and discrete probability distributions
Answer:-
Probability Distributions
A probability distribution describes how the values of a random variable are distributed. It gives the probability of each possible outcome and is divided into two main types:
- Discrete Probability Distribution
- Continuous Probability Distribution
1. Discrete Probability Distribution
Definition: A discrete probability distribution is used when the random variable can take only a finite or countably infinite number of values.
Characteristics:
- Probability is assigned to individual values.
- Uses the Probability Mass Function (PMF).
- The total probability is the sum of all individual probabilities:
∑ P(X = x) = 1
Example:
X = outcome of rolling a 6-sided die:
X ∈ {1, 2, 3, 4, 5, 6}
P(X = x) = 1/6 for all x
2. Continuous Probability Distribution
Definition: A continuous probability distribution is used when the random variable can take any value in an interval (uncountable real numbers).
Characteristics:
- Uses the Probability Density Function (PDF).
- Probability of exact value is zero: P(X = a) = 0
- Probability is defined over a range using integration:
P(a ≤ X ≤ b) = ∫ab f(x) dx
Total area under the PDF curve = 1 ∫-∞∞ f(x) dx = 1
Example:
X = height of people f(x) = probability density function for height P(160 ≤ X ≤ 170) = ∫160170 f(x) dx
Comparison Table
| Feature | Discrete Distribution | Continuous Distribution |
|---|---|---|
| Random Variable Type | Countable (e.g., 1, 2, 3) | Uncountable (e.g., real numbers) |
| Probability Tool | PMF (Probability Mass Function) | PDF (Probability Density Function) |
| P(X = a) | Non-zero | Zero |
| Total Probability | Sum of probabilities = 1 | Area under curve = 1 |
| Calculation Method | Summation | Integration |
| Example | Coin toss, die roll | Height, weight, temperature |
Conclusion
A discrete probability distribution is used when outcomes are countable, while a continuous probability distribution applies when outcomes are real-valued within a range. Both types are essential in modeling real-world randomness effectively.