Continuous Probability Distributions: Properties, Uniform and Normal Distributions

Continuous Probability Distributions

Continuous Probability Distributions and Their Properties

Continuous probability distributions describe the probabilities of outcomes for continuous random variables, which can take any value within a given interval. These distributions are fundamental in statistics for modeling real-world phenomena such as heights, weights, and measurement errors.

• Continuous Random Variable: A variable that can assume any value within an interval.

• Probability Density Function (PDF): A function, denoted as f(x), that describes the likelihood of a random variable taking a specific value. The probability that X falls between a and b is the area under the curve from a to b:

$P(a < X < b) = \int_a^b f(x) dx$

• Cumulative Distribution Function (CDF): The probability that X is less than or equal to x:

$F(x) = P(X \leq x) = \int_{-\infty}^x f(t) dt$

• For continuous variables, the probability at a single point is zero: $P(X = a) = 0$.

Examples from Biology and Medicine

• Drug Concentration: The amount of drug in the bloodstream after a dose is a continuous variable, often modeled with a normal distribution.

• Plant Heights: Heights of genetically identical plants can vary continuously due to environmental factors.

Mean, Variance, and Standard Deviation of a Continuous Random Variable

The mean (expected value) and variance for a continuous random variable are calculated using integrals:

• Mean (Expected Value):

$\mu = E[X] = \int_{-\infty}^{\infty} x f(x) dx$

• Variance:

$\sigma^2 = \int_{-\infty}^{\infty} (x - \mu)^2 f(x) dx$

Key Terms Table

Keyword	Definition
continuous random variable	A variable that can take any value in an interval; probabilities are assigned to ranges of values.
probability density function (PDF)	A non-negative function f(x) such that $P(a < X < b) = \int_a^b f(x) dx$ and the total area under the curve is 1.
cumulative distribution function (CDF)	The function $F(x) = P(X \leq x)$ giving the area under the PDF to the left of x.

The Uniform Distribution

Definition and Properties

The uniform distribution is the simplest continuous distribution, where all intervals of equal length within the range are equally likely. The PDF is constant between a and b:

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

• The area under the curve between a and b is 1.

• Expected Value (Mean):

$E[X] = \int_a^b x \frac{1}{b-a} dx = \frac{a+b}{2}$

• Variance:

$Var(X) = \frac{(b-a)^2}{12}$

• Standard Deviation:

$\sigma = \frac{b-a}{\sqrt{12}}$

• Probability between two points:

$P(c < X < d) = \int_c^d \frac{1}{b-a} dx = \frac{d-c}{b-a}$

Examples from Medicine and Biology

• Patient Arrival Times: If patients arrive randomly between 8 and 10 am, the time is uniformly distributed on [8, 10].

• Randomized Drug Administration: If a drug is administered at a random time between 2 and 3 pm, $T \sim Uniform(2, 3)$.

The Normal Distribution

Definition and Properties

The normal (Gaussian) distribution is the most widely used continuous distribution in statistics. It is symmetric, bell-shaped, and defined by its mean $\mu$ and standard deviation $\sigma$:

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

• Standard Normal Distribution: $N(0, 1)$, where $\mu = 0$ and $\sigma = 1$.

• Many biological and medical variables are approximately normal (e.g., blood pressure, height).

Finding Probability for a Normal Distribution

To find probabilities, standardize the variable using the z-score:

$z = \frac{x - \mu}{\sigma}$

• Use standard normal tables or software to find probabilities for $P(X > x)$ or $P(X < x)$.

Example: Antihypertensive Drug

• Suppose the reduction in systolic blood pressure after a drug follows $N(10, 6)$. What is the probability of a reduction of at least 15 mm Hg?

• Compute z: $z = \frac{15 - 10}{6} = 0.83$

• Find $P(X \geq 15) = P(Z \geq 0.83) = 0.2033$

Finding a Quantile for a Normal Distribution

Quantiles are values below which a certain percentage of data falls. For a given probability $p$, the quantile $x_p$ satisfies:

$F(x_p) = P(X \leq x_p) = p$

• Use the inverse CDF (quantile function) or standard normal tables to find quantiles.

Example: Therapeutic Drug Monitoring

• Suppose therapeutic blood drug concentrations follow $N(50, 10)$. To find the 97.5th percentile:

• Find $z_{0.975} = 1.96$

• Transform back: $x = 50 + 1.96 \times 10 = 69.6$

Key Terms Table

Keyword	Definition
normal distribution	A continuous, symmetric, bell-shaped distribution defined by mean $\mu$ and standard deviation $\sigma$.
standard normal distribution	A normal distribution with $\mu = 0$ and $\sigma = 1$.
quantile (percentile)	For a continuous distribution with CDF $F$, the value $x_p = F^{-1}(p)$ such that $P(X \leq x_p) = p$.

Summary

• Continuous probability distributions model variables that can take any value in an interval.

• The uniform and normal distributions are two key examples, each with specific properties and formulas.

• Probabilities are found as areas under the PDF curve; quantiles are found using the CDF or its inverse.