Probability Density Functions, Uniform Distribution, and the Normal Distribution

Probability Density Functions and Continuous Random Variables

Probability Density Function (PDF)

A probability density function (pdf) is a mathematical function used to describe the probability distribution of a continuous random variable. It must satisfy two key properties:

• Total Area: The total area under the graph of the pdf over all possible values of the random variable must equal 1.

• Non-Negativity: The height of the graph must be greater than or equal to 0 for all possible values of the random variable.

For continuous random variables, the probability of observing any single, specific value is zero. Instead, probabilities are calculated for intervals of values.

Key Points

• The area under the graph of the pdf over an interval represents the probability that the random variable falls within that interval.

• The pdf provides the probability density at a point, not the probability of the variable taking that exact value.

• Continuous random variables have uncountably infinite possible values.

Example: Uniform Distribution

Suppose the random variable X represents the time (in minutes) a friend is late, with all times between 0 and 30 minutes equally likely. The pdf for X is constant over the interval [0, 30]:

• For 0 ≤ x ≤ 30, the pdf is $f(x) = \frac{1}{30}$

To find the probability that your friend is between 10 and 20 minutes late:

• Width of interval: 10 minutes

• Probability: $10 \times \frac{1}{30} = \frac{1}{3}$

To find the interval corresponding to a 20% probability:

• Set $x_0 \times \frac{1}{30} = 0.2$

• Solve: $x_0 = 30 \times 0.2 = 6$

• There is a 20% probability your friend will arrive within the next 6 minutes.

Discrete vs. Continuous Distributions

PMF vs PDF

Discrete random variables use a probability mass function (pmf), while continuous random variables use a probability density function (pdf).

• PMF: Assigns probabilities to specific values (e.g., binomial distribution).

• PDF: Assigns probability densities; probabilities are found by integrating over intervals (e.g., normal distribution).

Uniform Distribution

Definition and Properties

A uniform distribution is a continuous probability distribution where all intervals of equal length within the distribution's range are equally likely.

• The graph of the uniform distribution is a rectangle.

• For X uniformly distributed over [a, b], the pdf is $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$.

Normal Distribution

Definition and Features

The normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is defined by two parameters: the mean (μ) and the standard deviation (σ).

• Symmetric about its mean (μ).

• Mean = median = mode; single peak at x = μ.

• Inflection points at x = μ - σ and x = μ + σ.

• Total area under the curve is 1.

• As x increases or decreases without bound, the curve approaches but never touches the horizontal axis.

Normal Probability Density Function

The pdf of the normal distribution is:

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

where μ is the mean and σ is the standard deviation.

Empirical Rule (68-95-99.7 Rule)

• Approximately 68% of the area is within one standard deviation of the mean: $x = \mu \pm \sigma$

• Approximately 95% is within two standard deviations: $x = \mu \pm 2\sigma$

• Approximately 99.7% is within three standard deviations: $x = \mu \pm 3\sigma$

Inflection Points

Inflection points are where the curvature of the normal curve changes direction, located at $x = \mu - \sigma$ and $x = \mu + \sigma$.

Family of Normal Curves

• Changing μ shifts the curve horizontally.

• Changing σ alters the spread (flatter for larger σ, steeper for smaller σ).

Median and Mean of a Density Curve

• The median is the point dividing the area under the curve in half.

• The mean is the balance point of the curve.

• For symmetric curves, mean and median coincide; for skewed curves, the mean is pulled toward the long tail.

Applications and Examples

Area Under the Normal Curve

The area under the normal curve for an interval represents:

• The proportion of the population with the characteristic described by the interval.

• The probability that a randomly selected individual has the characteristic described by the interval.

Example: Cholesterol Levels

Suppose serum cholesterol for males aged 20-29 is normally distributed with $\mu = 180$ mg/dL and $\sigma = 36.2$ mg/dL.

• Individuals with cholesterol > 200 mg/dL are considered to have high cholesterol.

• If the area under the curve to the right of $x = 200$ is 0.2903, then:

  • 29.03% of males in this age group have high cholesterol.

  • The probability that a randomly selected male has high cholesterol is 0.2903.

Example: Heights of Three-Year-Old Females

Given a mean height $\mu = 38.72$ inches and standard deviation $\sigma = 3.17$ inches, the distribution of heights is approximately normal. The area under the curve for a given interval (e.g., 40 to 40.9 inches) closely matches the proportion of individuals in that interval.

Height (inches)	Relative Frequency
29.0–29.9	0.001
30.0–30.9	0.001
31.0–31.9	0.003
32.0–32.9	0.005
33.0–33.9	0.011
34.0–34.9	0.017
35.0–35.9	0.027
36.0–36.9	0.051
37.0–37.9	0.117
38.0–38.9	0.170
39.0–39.9	0.183
40.0–40.9	0.151
41.0–41.9	0.112
42.0–42.9	0.077
43.0–43.9	0.044
44.0–44.9	0.024
45.0–45.9	0.011
46.0–46.9	0.003
47.0–47.9	0.001

Summary Table: Uniform vs Normal Distribution

Feature	Uniform Distribution	Normal Distribution
Shape	Rectangle	Bell-shaped
Parameters	Lower and upper bounds (a, b)	Mean (μ), Standard deviation (σ)
PDF	$f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$	$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
Symmetry	Yes	Yes
Inflection Points	None	$x = \mu - \sigma$, $x = \mu + \sigma$

Key Formulas

• Uniform PDF: $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$

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

Additional info:

• These notes cover material relevant to Chapter 6 (Discrete Probability Distributions) and Chapter 7 (The Normal Probability Distribution) of a college statistics course.

• Examples and tables have been expanded for clarity and completeness.