Continuous and Normal Probability Distributions

Continuous Random Variables

Definition and Examples

Continuous random variables are variables that can take infinitely many possible values within a given range. Unlike discrete random variables, which have countable outcomes, continuous random variables are measured rather than counted.

• Definition: A continuous random variable is a variable that can assume any value within an interval.

• Examples:

  • Time of arrival

  • Numerical grades

Probability of Continuous Random Variables

Interval Probabilities

For a continuous random variable X, we are interested in the probability that X falls within a certain interval, rather than at a specific value. The probability that X equals any particular value is always zero:

• Probability at a point: $P(X = c) = 0$ for any specific value c.

• Probability in an interval: $P(a < X < b)$ is the probability that X falls between a and b.

Continuous Probability Function

Uniform Probability Distribution

A continuous probability distribution describes the likelihood of all possible values of a continuous random variable. The uniform distribution is a simple example where all intervals of equal length within the range are equally likely.

• Example: If a package can arrive at any time between 0 and 60 minutes, the probability of arrival in any 1-minute interval is the same.

• Uniform distribution: $X$ is said to follow a uniform distribution if every interval of equal length within the range has the same probability.

Graphs in Continuous Random Variables

Representing Distributions

Unlike discrete random variables, continuous random variables cannot be represented by tables. Instead, their distributions are described using mathematical formulas and graphed as curves or shapes.

• Probability density function (pdf): Used to represent the distribution of continuous random variables.

Probability Density Function (pdf)

Properties of the PDF

The probability density function (pdf) is an equation that describes the distribution of a continuous random variable. It must satisfy two key properties:

• Total area under the curve: The area under the graph of the pdf over all possible values must equal 1.

• Non-negativity: The graph of the pdf must be greater than or equal to zero for all possible values.

Uniform Density Function Graph

Example and Interpretation

The graph of a uniform density function is a rectangle, where the area represents the total probability (which is 1). The function value is zero outside the interval.

• Example: For $X$ between 0 and 60, the pdf is constant within the interval and zero elsewhere.

• Interpretation: The probability that $X$ falls within any subinterval is proportional to the length of that interval.

Calculating Probabilities

Interval Probabilities in Uniform Distribution

To find the probability that a continuous random variable falls within a specific interval, calculate the area under the pdf over that interval.

• Example: Probability that a package arrives between 10:15 am and 10:30 am (a 15-minute interval out of 60 minutes): $P = \frac{15}{60} = 0.25$ or 25% chance.

Normal Probability Distribution

Introduction to the Normal Distribution

Many continuous random variables, such as biological measurements or IQ scores, follow a bell-shaped curve known as the normal distribution.

• Normal distribution: A symmetric, bell-shaped probability distribution characterized by its mean ($\mu$) and standard deviation ($\sigma$).

• Example: IQ scores can be modeled by a normal distribution with $\mu = 100$ and $\sigma = 15$.

Properties of the Normal Distribution

Key Characteristics

• Symmetry: The curve is symmetric about its mean ($\mu$).

• Mean, median, mode: All are equal and located at the highest point of the curve.

• Inflection points: Occur at $\mu - \sigma$ and $\mu + \sigma$.

• Total area: The area under the curve is 1.

• Half-area: The area to the left or right of the mean is 0.5.

The Empirical Rule

Area Under the Normal Curve

The empirical rule describes the proportion of data within certain intervals around the mean in a normal distribution:

• Approximately 68% of the area is within one standard deviation: $\mu - \sigma < X < \mu + \sigma$

• Approximately 95% is within two standard deviations: $\mu - 2\sigma < X < \mu + 2\sigma$

• Approximately 99.7% is within three standard deviations: $\mu - 3\sigma < X < \mu + 3\sigma$

Standard Normal Distribution and Z-Scores

Standardization

The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. Any normal variable can be converted to a standard normal variable (Z-score) using:

• Z-score formula: $Z = \frac{X - \mu}{\sigma}$

• Application: Allows comparison across different normal distributions and use of standard normal tables.

Using the Standard Normal Table

Finding Probabilities

The standard normal table provides the area (probability) to the left of a given Z-score. To find the probability to the right, subtract the table value from 1.

• Example: For $z = 1.33$, area to the left is 0.9082, so area to the right is $1 - 0.9082 = 0.0918$.

Applications of the Normal Distribution

Percentiles and Real-World Examples

Normal distributions are used to find percentiles and probabilities in real-world contexts, such as cholesterol levels or test scores.

• Example: If total cholesterol for males aged 20-29 is normally distributed with $\mu = 180$ mg/dL and $\sigma = 36.2$ mg/dL, the probability of having high cholesterol (greater than 200 mg/dL) can be found by calculating the area under the curve to the right of 200.

Summary Table: Properties of Distributions

Distribution	Shape	Parameters	Key Properties
Uniform	Rectangle	Interval [a, b]	Equal probability for all intervals of equal length
Normal	Bell-shaped	Mean ($\mu$), Std. Dev. ($\sigma$)	Symmetric, mean=median=mode, empirical rule applies

Additional info: These notes expand on the provided slides by including formal definitions, formulas, and examples for continuous and normal probability distributions, as well as a summary table for comparison.