Module 7

Probability Distributions

Introduction

Probability distributions are foundational concepts in statistics, describing how probabilities are assigned to the outcomes of random experiments. Understanding these distributions is essential for statistical inference, data analysis, and real-world problem-solving in health sciences and beyond. This module covers both discrete and continuous probability distributions, including their properties, calculations, and applications.

Key Concepts and Learning Objectives

• Define and distinguish between discrete and continuous probability distributions.

• Calculate probabilities using probability mass functions (PMFs) and probability density functions (PDFs), as well as cumulative distribution functions (CDFs).

• Describe the characteristics of key distributions: Uniform, Binomial, Poisson, and Normal.

• Compute expected values (means) and variances for given distributions.

• Apply probability distributions to solve real-world problems, especially in health sciences.

• Interpret graphs and tables related to probability distributions.

• Understand the importance of the Normal distribution in statistical inference.

Random Experiments, Sample Space, and Random Variables

Random Experiment and Sample Space

• Random Experiment: An action or process that leads to one of several possible outcomes, e.g., tossing a coin twice.

• Sample Space (S): The set of all possible outcomes. For tossing a coin twice: S = {HH, HT, TH, TT}.

• Sample Point: An individual outcome in the sample space.

Random Variables (RV)

• A random variable (X) assigns a real number to each outcome in the sample space.

• Notation: Use uppercase letters (X, Y) for random variables; lowercase (x, y) for their values.

• Example: Let X = number of tails in two coin tosses. Possible values: X = 0, 1, 2.

Probability Distributions

Definition and Representation

• A probability distribution describes how probabilities are distributed over the possible values of a random variable.

• Can be represented as a list, table, graph, or formula.

• The sum of all probabilities for a discrete random variable is 1.

Types of Random Variables

• Discrete Random Variable: Takes countable values (e.g., number of patients, number of side effects).

• Continuous Random Variable: Takes any real value within an interval (e.g., body temperature, blood pressure).

• Binary Variable: Only two possible outcomes (e.g., positive/negative, recovered/not recovered).

• Categorical Variable: Distinct categories with no inherent order (e.g., blood type, treatment type).

Probability Mass Function (PMF) and Probability Density Function (PDF)

PMF (Discrete Random Variables)

• The probability mass function (PMF) gives the probability that a discrete random variable is exactly equal to some value x: $P(X = x)$.

• Example: Tossing a coin twice, X = number of tails.

PDF (Continuous Random Variables)

• The probability density function (PDF) describes the likelihood of a continuous random variable taking a value in an interval.

• Probability at a single point is zero: $P(X = x) = 0$.

• Probability over an interval: $P(a < X < b) = \int_a^b f(x) dx$.

Comparison Table: PMF vs PDF

Type	Function	Probability at a Point	Computation
Discrete	PMF	$P(X = x) > 0$	Summation
Continuous	PDF	$P(X = x) = 0$	Integration (area under curve)

Expectation (Mean) and Variance

• Expected Value (Mean): $E(X) = \sum x \cdot p(x)$ for discrete, $E(X) = \int x f(x) dx$ for continuous.

• Variance: $Var(X) = E[(X - \mu)^2] = \sum (x - \mu)^2 p(x)$ (discrete), $Var(X) = \int (x - \mu)^2 f(x) dx$ (continuous).

• Standard Deviation: $\sigma = \sqrt{Var(X)}$

Common Probability Distributions

Uniform Distribution

Discrete Uniform

• All outcomes are equally likely.

• PMF: $P(X = x_i) = \frac{1}{k}$ for $i = 1, 2, ..., k$

• Example: Drawing a ball from a box with 3 colors, each color has probability 1/3.

Continuous Uniform

• All values in interval [A, B] are equally likely.

• PDF: $f(x) = \frac{1}{B - A}$ for $A \leq x \leq B$; $f(x) = 0$ otherwise.

• Mean: $\mu = \frac{A + B}{2}$

• Variance: $\sigma^2 = \frac{(B - A)^2}{12}$

• Example: IV infusion time uniformly distributed between 30 and 60 minutes.

Binomial Distribution

• Models the number of successes in n independent Bernoulli trials (each with probability p of success).

• PMF: $P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$ for $x = 0, 1, ..., n$

• Mean: $\mu = np$

• Variance: $\sigma^2 = np(1-p)$

• Assumptions: Fixed number of trials, independent trials, constant probability of success.

• Example: Probability that exactly 8 out of 10 patients recover if treatment is 80% effective.

Poisson Distribution

• Models the number of times an event occurs in a fixed interval of time or space, given a constant mean rate $\lambda$.

• PMF: $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$ for $k = 0, 1, 2, ...$

• Mean and Variance: $\mu = \sigma^2 = \lambda$

• Assumptions: Events occur independently, at a constant rate, and two events cannot occur at exactly the same instant.

• Example: Number of patients arriving at a clinic per hour.

Normal Distribution

• Continuous, symmetric, bell-shaped distribution; also called the Gaussian distribution.

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

• Mean = median = mode = $\mu$

• Standard deviation $\sigma$ determines the spread.

• Empirical Rule: 68.2% within 1$\sigma$, 95.4% within 2$\sigma$, 99.7% within 3$\sigma$ of the mean.

• Standard Normal Distribution: $Z = \frac{X - \mu}{\sigma}$

• Applications: Blood pressure, body temperature, BMI, test scores, etc.

Normal Approximation

• Binomial and Poisson distributions can be approximated by the normal distribution when n is large and p is not too close to 0 or 1.

• Continuity correction: For discrete to continuous, use $P(X < k) \approx P(Y < k + 0.5)$.

Cumulative Distribution Function (CDF)

• The CDF gives the probability that a random variable X is less than or equal to a value x: $F(x) = P(X \leq x)$.

• For continuous distributions, $F(x) = \int_{-\infty}^x f(t) dt$.

• Used to find percentiles, probabilities over intervals, and to interpret Z-tables.

Using Z-Tables and Calculating Probabilities

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

• Use Z-tables to find probabilities and percentiles for normal distributions.

• Example: If SBP is normally distributed with $\mu = 120$ mmHg, $\sigma = 15$ mmHg, the probability that SBP > 140 mmHg is $P(Z > 1.33) = 1 - 0.9082 = 0.0918$.

Applications in Health Sciences

• Vital signs (blood pressure, temperature, heart rate) often follow normal distributions.

• Drug response, side effects, and dosage studies use probability distributions for analysis.

• Resource allocation, scheduling, and quality control in healthcare use uniform and Poisson distributions.

• Clinical trials and diagnostic test interpretation rely on binomial and normal distributions.

Summary Table: Key Distributions

Distribution	Type	Parameters	Mean	Variance	Example
Uniform (Discrete)	Discrete	k (number of outcomes)	$\frac{k+1}{2}$	$\frac{(k^2-1)}{12}$	Rolling a fair die
Uniform (Continuous)	Continuous	A, B (interval)	$\frac{A+B}{2}$	$\frac{(B-A)^2}{12}$	IV infusion time
Binomial	Discrete	n, p	$np$	$np(1-p)$	Number of recoveries
Poisson	Discrete	$\lambda$	$\lambda$	$\lambda$	Clinic arrivals/hour
Normal	Continuous	$\mu$, $\sigma$	$\mu$	$\sigma^2$	Blood pressure

Conclusion

Probability distributions are essential tools in statistics, enabling the modeling and analysis of random phenomena in health sciences. Mastery of these concepts allows for effective data interpretation, decision-making, and research in medical and scientific contexts.