Continuous Probability Distribution

Study Notes

Continuous probability refers to a type of probability distribution where the outcomes are continuous and can take on any value within a certain range or interval.
It is used to model random variables that can take on any value within a certain range, such as heights, weights, and temperatures.

Continuous probability distributions are characterized by the following properties:
- The probability of a single point is zero (i.e., P(x = a) = 0)
- The probability of a range of values is greater than zero (i.e., P(a ≤ x ≤ b) > 0)
- The probability distribution is continuous, meaning that there are no gaps or jumps in the distribution

Uniform Distribution:
- A continuous distribution where every possible value within a given range has an equal probability of occurring
- Examples: rolling a die, drawing a random number between 0 and 1
Normal Distribution (Gaussian Distribution):
- A continuous distribution that is symmetric and bell-shaped
- Examples: heights of people, IQ scores, errors in measurement
Exponential Distribution:
- A continuous distribution that models the time between events in a Poisson process
- Examples: time between arrivals, time between failures

A PDF is a function that describes the probability distribution of a continuous random variable
The PDF is non-negative and integrates to 1 over the entire range of the variable
The PDF is used to calculate probabilities of ranges of values, such as P(a ≤ x ≤ b)

A CDF is a function that describes the cumulative probability of a continuous random variable
The CDF is a monotonically increasing function that ranges from 0 to 1
The CDF is used to calculate probabilities of ranges of values, such as P(x ≤ a)

Continuous probability is used in a wide range of fields, including:
- Engineering: modeling errors, signal processing, and quality control
- Economics: modeling stock prices, returns, and portfolio optimization
- Medicine: modeling patient outcomes, treatment effects, and disease progression

Refers to a type of probability distribution where outcomes are continuous and can take on any value within a certain range or interval.
Used to model random variables that can take on any value within a certain range, such as heights, weights, and temperatures.

The probability of a single point is zero.
The probability of a range of values is greater than zero.
The probability distribution is continuous, meaning there are no gaps or jumps in the distribution.

A continuous distribution where every possible value within a given range has an equal probability of occurring.
Examples: rolling a die, drawing a random number between 0 and 1.

A continuous distribution that models the time between events in a Poisson process.
Examples: time between arrivals, time between failures.

A function that describes the probability distribution of a continuous random variable.
The PDF is non-negative and integrates to 1 over the entire range of the variable.
Used to calculate probabilities of ranges of values, such as P(a ≤ x ≤ b).

A function that describes the cumulative probability of a continuous random variable.
A monotonically increasing function that ranges from 0 to 1.
Used to calculate probabilities of ranges of values, such as P(x ≤ a).

Used in engineering to model errors, signal processing, and quality control.
Used in economics to model stock prices, returns, and portfolio optimization.
Used in medicine to model patient outcomes, treatment effects, and disease progression.
Used in a wide range of fields.