Explain continuous and discrete probability distributions

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

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.